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«* • 







PRACTICAL ADVANCED 
NAVIGATION 


BY THE SAME AUTHOR 


SIMPLE RULES AND PROBLEMS 
IN NAVIGATION 

Corrected and Revised 

By BRADLEY JONES 

Instructor of Navigation 
Massachusetts Institute of Technology 


Contains all the rules, with various 
problems worked out in full, of all the 
old and new methods of finding a ship’s 
position at sea, accompanied by a set of 
answers to the problems for the current 
year; also gives the International Rules 
of the Road. 


E. P. DUTTON & COMPANY 






1 

PRACTICAL ADVANCED 
NAVIGATION 


BY 

CHARLES H. CUGLE 


y 


AUTHOR OF 

“Simple Rules and Problems in Navigation’* 


Formerly Chief of Gulf Section, U. S. Shipping Board Navigation School* 
and U. S. Assistant Inspector of Hulls, Steamboat Inspection 
Service, New Orleans, La, 



NEW YORK 

E. P. DUTTON & COMPANY 

681 Fifth Avknue 






Copyright, 1922 j 

By E. P. DUTTON & COMPANY 


AU right* reterved 


y\<ss^ 

• C 


©CI.A6bG538 's^ 



Printed in the United State* of Amtrica 


OCT 30 *22 


'9 



TO 

Hon* (Sear^e Hljler 

SUPEBVISINQ IN8PECTOR-GENEKAL, U. 8. STEAMBOAT 
IN8PECTION SERVICE, WASHINGTON, D, C. 

WHOSE LIFE HAS BEEN DEVOTED TO THE BETTERMENT 
OF THE AMERICAN MERCHANT MARINE OFFICER, 


THIS WORK IS RESPECTFULLT DEDICATED, 


ACKNOWLEDGMENT 


The author wishes to extend his thanks to Captain 
John T. Sullivan, Master S/S ‘'West Kasson,” U. S. 
Shipping Board, for his kind co-operation in compiling 
the information contained herein. 



PREFACE 


'^Simple Rules and Problems in Navigations^ having 
met with such favorable comment and success, it has 
occurred to me that a volume containing simple rules and 
problems in the practical higher branches of navigation 
would also be received with the same spirit. 

The subjects taken up here have, more or less, been 
clouded with a lot of mystery, whereas, each one of them 
is as simple as any other problem in navigation, when 
given with rules that the ordinary sea-going man can under¬ 
stand. This is especially true of Great Circle Sailing. 

The subjects covered are ‘^Deviation by Star, Planet, 
and Moon,^^ ^^Construction of a Mercator Chart,^^ Great 
Circle Sailing,’^ “Double Chronometer Method of Deter¬ 
mining Ship’s Position (commonly known as Johnson’s 
Method, but in this volume worked entirely from Bow- 
ditch),” “Tangent Method of Determining Ship’s Position 
by Sumner Lines,” “Marcq St. Hilaire Method,” “Chart- 
lets with the Marcq St. Hilaire and Siunner problems 
plotted on same.” 

There are given two methods of determining Great 
Circle Courses, and three formulas for working Marcq St. 
Hilaire method. 

The books used in working the problems are “Bowditch’s 
American Practical Navigator,” “1922 American Nautical 
Almanac,” “American Azimuth Table H. O. No. 71,” 
“Azimuths of Celestial Bodies H. 0. No. 120.” 

The Great Circle Courses are worked out complete for 
a Trans-Atlantic voyage in Example 1. 

Three of the problems in Double Chronometer method 
are also worked out in Marcq St. Hilaire (3 formulas) and 
the position lines are plotted on Chartlets, where it can be 
seen that they agree. Two of the problems in Sumner 
method are also taken from Examples 1 and 5 of Double 
Chronometer and plotted on Chartlets. 

The rules have been given in as plain language as pos¬ 
sible, and no attempt has been made to go into theory. 

vii 



viii 


CONTENTS 


There are many excellent books on the theory, but very few 
that the average sailor can understand. If the reader 
wishes to delve more deeply into the reasons, he is respect¬ 
fully referred to Bowditch^s ^^American Navigator,^' which, 
in the Author^s opinion, cannot be improved upon. 

It is hoped that this volume will be of help, and meet 
with your approval. 

Charles H. Cugle. 

1622 Pine Street, 

New Orleans, La. 


CONTENTS 


CHAPTER I 

PAOB 

Deviation by Star, Moon, and Planet.1 

CHAPTER II 

Great Circle Sailing.4 

CHAPTER III 

Double Chronometer Method of Determining by Computation 
THE Position of Vessel by Two Observations .... 21 

CHAPTER IV 

Construction of a Mercator Chart.34 

CHAPTER V 

Sumner Lines by Tangent Method.38 

CHAPTER VI 

Marcq St, Hilaire Method.43 

Extracts from the American Nautical Almanac for the Year 
1922 52 


ix 









r 




♦ 




) 

\ 

7 



PRACTICAL ADVANCED 
NAVIGATION 




« 




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f 


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PRACTICAL ADVANCED NAVIGATION 


CHAPTER I 

DEVIATION BY STAR, MOON AND PLANET 

The method of determining deviation by celestial bodies 
other than the sun becomes imperative in certain localities 
where variation is changing quickly and climatic conditions 
are such that observations of the sun are not available at all 
times, also when navigating in dangerous waters. There¬ 
fore this method should be thoroughly understood by all 
navigators. 

Rule : 

1. Correct chronometer to G. M. T. 

2. From the G. M. T. subtract the Longitude in Time 

when West and add it to the G. M. T. when 
Longitude is East. Result will be Local Mean 
Time (L. M. T.). 

3. Add to the L. M. T. the Sun^s Right Ascension 

(S. R. A.) for Greenwich Date and correction for 
G. M. T. Result will be Local Siderial Time 
(L. S. T.). 

4. From the L. S. T. subtract the correct Right Ascen¬ 

sion of the Body observed. Result will be Body^s 
Hour Angle (H. A.). 

5. Enter Azimuth Table with Latitude to nearest 

degree, declination at top and H. A. at side in the 
P. M. column, and take out Azimuth or True 
Bearing. 

Note. If the declination exceeds 23° it is necessary 
to use Azimuths of Celestial Bodies^^ H. O. 120 
from declination 24° to 70°, Lat. 0° to 70°. In 
using H. 0. 120 when the Lat. and Deck are of 



2 


PRACTICAL ADVANCED NAVIGATION 


same name, the Azimuth is read from N. to E. or 
from N. to W. in North Lat. and from S. to E. and S. 
to W. in South Lat. 

When Lat. and Decl. are contrary names, it is neces¬ 
sary to enter the table with the Supplement of the 
Hour Angle. For example: H. A. 7h 50m. Supple¬ 
ment is 12h—7h 50m = 4h 10m which is H. A. to 
enter table. The Supplement of the Azimuth found 
with this H. A. and Decl. is the True Bearing or Azi¬ 
muth. For example: Lat. 37° N. Decl. 28° S. H. A. 
8h 20m. Body West = Supplement of H. A. 
8h 20m =H. A. 3h 40m. Azimuth Lat. 37° N. 
Decl. 28° S. H. A. 3h 40m is 84° 28'. 180° - 

84° 28'=N. 95° 32' W. True Bearing. 

6. The True Bearing is now applied to the Compass 
Bearing, and the Error of Compass and Devia¬ 
tion is determined by same rule as in Azimuth by 
the Sun, 


EXAMPLE 1 

Aug. 10, 1922. A. M. at ship. Compass bearing Star 
Fomalhaut N. 86° W. Lat. 37° 10' N. Long. 60° W. 
Chronometer read 8h 18m. Variation 10° W. Required 
Error and Deviation of Compass? 

G. M. T. 9d 20h 18m 00s * Decl. 30° 2' S. 

Lon. in T. 4 00 00 


L. M. T. 16 18 00 

S.R.A. 9d 9 8 47 Az.Lat.37°N. Decl.30°S. H. A. 2h 37m 


COIT. 


3 

20 




L. S. T. 

25 

30 

07 

Az. from Table 

90° 

26' 

*R. A. 

22 

53 

24 

As Diff. names 

180 

00 

*H. A. 

2 

36 

43 

True Brg. 

N 89 

34 W 





Comp. Brg. 

N 86 

00 W 





Comp. Error 

3 

34 W 





Variation 

10 

00 W 


Deviation 


6 26E 








PRACTICAL ADVANCED NAVIGATION 


3 


EXAMPLE 2 

June 5, 1922. P. M. at ship. Compass bearing Star 
Antares S. 70° E. Lat. 30° N. Long. 45° W. Chronom¬ 
eter read 9h 45m. Variation 16° W. Required Error and 
Deviation of Compass? 


G.M.T. 5d 

9h 

45m 

00s 

♦ Decl. 26° 

’ 15' S. 



Lon. in T. 

3 

00 

00 





L. M. T. 

6 

45 

00 





S. R. A. 5d 

4 

52 

31 

Az.Lat.30° N. 

Decl. 26° S. H.A.4h46m 

Corr. 


1 

36 





L. S. T. 

11 

39 

07 

True Brg. 

N105° 

30' 

E 

♦R. A. 

16 

24 

40 

Comp. Brg. 

NllO 

00 

E 

*H. A. 

4 

45 

33 

Comp. Error 

4 

30 

W 





Variation 

16 

00 

W 


Deviation 11 30 E 


EXAMPLE 3 

March 9, 1922. P. M. at ship. Compass bearing of 
Moon, East. Variation 9° W. Lat. 37° N. Long. 64° W. 
Chronometer read 9h 6m. Required Error and Deviation of 
Compass? 


G. M. T. 9d 

9h 6m 

00s 

Decl. 9d 8h 15° 46'.8 N 

Lon. in T. 

4 

16 

00 

Corr. Table IV - 6.6 

L. M. T. 

4 

50 

00 

True Decl. 15 40 .2 N 

S. R. A. 9d 

23 

5 

34 


Corr. 


1 

30 

R. A. 9d 8h 7h 54m 06s 





Corr. Table IV + 2 44 

L. S. T. 

27 

57 

04 


Moon's R. A. 

7 

56 

50 

Correct R. A. 7 56 50 


20 

00 

14 



24 

00 

00 

Az. Lat. 37° N. Decl. 16° N. 


Moon^sH.A. 3 59 46 

True Brg. N 94° 45' E 
Comp. Brg. N 90 00 E 


Comp. Error 4 45 E 
Variation 9 00 W 


Deviation 


13 45 E 















CHAPTER II 


GREAT CIRCLE SAILING 

In this chapter are given two methods of determining 
Great Circle Courses, namely, by Time Azimuth and Com¬ 
putation of Logs. 

By using either one of these methods it is only necessary 
to use the chart in order to determine whether the great 
circle will lead across land or in dangerous waters. In such 
a case a certain position should be fixed on as a limit for the 
great circle, and then another great circle from that position 
to point of destination. This method of more than one great 
circle is called Composite Sailing. 

The examples given below will explain themselves. 


EXAMPLE 1 
(Time Azimuth Method) 

Required great circle courses and great circle distance 
from a point off Cape Hatteras in Lat. 35° N., Long. 73° 
W. to a point off Cape St. Vincent in Lat. 37° N. Long, 
9° W. 

Note: A ship when sailing on a great circle course is 
always heading for her port of destination, while when sail¬ 
ing on a Mercator course she is never headed for her destina¬ 
tion until at the end of her voyage. In order to keep on the 
great chcle track it is necessary to alter the course at various 
intervals. As the difference of longitude decreases in 
approaching destination, so also will the hour angle, and as 
the hour angle decreases the greater will be the change in 
Azimuth. Therefore, in determining the distances to be 
run on each course, the change of azimuth will have to be 
taken into consideration, and the distance on each course 
reduced accordingly. 


T 


PRACTICAL ADVANCED NAVIGATION 


5 


TO FIND GREAT CIRCLE COURSE 

Rule: 

With the Latitude of Departure (35°) as the Lat. 

With the Latitude of Destination (37°) as the Declina¬ 
tion (see note). 

With the Difference of Longitude between departure 
and destination, converted into time, as the hour 
angle. 

Enter the Azimuth Table and the Azimuth obtained 
will be the initial or First (true) Course. 

Note: The Latitude of the Destination as the Declina¬ 
tion will remain constant for all courses. 

For Example: 

Lat. Dep. 35° N Long. Dep. 73° W 
Long. Dest. 9° W 

Diff. Long. 64° = H. A. 4h 16m 

Enter Az. Table at Lat. 35° Decl. 37° H. A. 4h 16m= 
Az. 67° 51' or 68°. 

Initial or First (true) Course is 68°. 

As the change in the Azimuth is small on the First Course 
we will assume the distance run on this course to be 100 
miles. 

Entering Table 2 (Bowditch) we find for Course 68°, 
Distance 100, Diff. Lat. 37.5, Dep. 92.7. Diff. Long. 113. 

Lat. left 35° 00' 00" N Long, left 73° 00' 00" W 
Diff. Lat. 37 30 N Diff. Long. 1 53 00 E 

Lat. In 35 37 30 N Long. In 71 07 00 W=Dep.for2dG. 

2d Course 

Lat. Dep. 35° 37'.5 Long. Dep. 71° 07' W 

Long. Dest. 9 00 W 

Diff. Long. 62 07 =H. A. 4h 8m.5 

Azimuth Lat. 35° 37'.5 Dec. 37°. H. A. 4h 8m.5 =2d Course 69° 4' or 69°. 
Table 2 at 69°. Dist. 100 =Diff. Lat. 35.8. Dep. 93.4. Diff. Long. 115^5i 

Lat. Left 35° 37' 30" N Long. Left 71° 07' 00" W 
Diff. Lat. 35 48 N Diff. Long. 1 55 30 E 

Lat. In 36 13 18 N Long. In 69 11 30 W=Dep. for 3d C. 








6 


PRACTICAL ADVANCED NAVIGATION 


3d Course 

Lat. Dep. 36° 13'.3 Long. Dep. 69° 11'.5 W 

Long. Dost. 9 W 


Diff. Long. 60 11'.5 =H. A. 4h Om.8 

Azimuth Lat. 36° 13'.3 Dec. 37°. H. A. 4h Om.8 =3d Course 70° 6' or 70°. 
Table 2 at 70°. Dist. 100 = Diff. Lat. 34.2. Dep. 94.0. Diff. Long. 117. 

Lat. Left 36° 13' 18" N Long. Left 69° 11' 30" W 
Diff. Lat. 34 12 N Diff. Long. 1 57 E 


Lat. In 36 47 30 N Long. In 67 14 30 W=Dep.for4thC. 


4ih Course 

Lat. Dep. 36° 47'.5 Long. Dep. 67° 14'.5 W 

Long. Dest. 9 W 

Diff. Long. 58 14 .5 =H. A. 3h 53m 

Azimuth Lat. 36° 47'.5. Dec. 37°. H.A. 3h 53m =4th Course 71° 14' or 71°. 
Table 2 at 71°. Dist. 100= Diff. Lat. 32.6. Dep. 94.6. Diff. Long. 118.5. 

Lat. Dep. 36° 47' 30" N Long. Dep. 67° 14' 30" W 

Diff. Lat. 32 36 N Diff. Long. 1 58 30 E 

Lat. In 37 20 06 N Long. In 65 16 00 W=Dep. for 5th G. 


6 th Course 

Lat. Dep. 37° 20'.1 Long. Dep. 65° 16' W 

Long. Dest. 9 W 

Diff. Long. 56 16 =H. A. 3h 45m.l 

Azimuth Lat. 37° 20'. 1 Decl. 37°. H. A. 3h 45m.1 = 5th Course 72° 29'. 
Table 2 at 72° 29'. Dist. 100 =Diff. Lat. 30. Dep. 95.3. Diff. Long. 120. 

Lat. Left 37° 20' 06" N Long. Left. 65° 16' W 

Diff. Lat. 30 00 N Diff. Long. 2 00' E 

Lat. In 37 50 06 N Long. In 63 16 W =Dep. for 6th G. 


6th Course 

Lat. Dep. 37° 50'.1 Long. Dep. 63° 16' W 

Long. Dest. 9 W 


Diff. Long. 54 16 =H. A. 3h 37m.l 

Azimuth Lat. 37° 50'.1 Dec. 37°. H. A. 3h 37m.l =6th C. 73° 44' or 74°. 
Table 2 at 74°. Dist. 100 = Diff. Lat. 27.6. Dep. 96.1. Diff. Long. 122. 

Lat. Left 37° 50' 06" N Long. Left 63° 16' W 
Diff. Lat. 27 36 N Diff. Long. 2 02 E 


Lat. In. 38 17 42 N Long. In. 61 14 W= Dep. for 7th G. 














PRACTICAL ADVANCED NAVIGATION 


7 


7th Course 

Lat. Dep. 38° 17'.7 Long. Dep. 61° 14' W 

Long. Dost. 9 W 


Diff. Long. 52 14 =H. A. 3h 28m.9 

Azimuth Lat. 38° 17'.7. Dec. 37°. H. A. 3h 28m.9 =7th Course 75°. 
Table 2 at 75°. Dist. 100 =Diff. Lat. 25.9. Dep. 96.6 Diff. Long. 123. 

Lat. Left 38° 17' 42" N Long. Left 61° 14'W 
Diff. Lat. 25 54 N Diff. Long. 2 3'E 


Lat. In 38 43 36 N Long. In 59 11 W=Dep. for 8th G. 


8th Course 

Lat. Dep. 38° 43'.6 Long. Dep. 59° 11' W 

Long. Dest. 9 W 

Diff. Long. 50 11 =H. A. 3h 20m.7 

Azimuth Lat. 38° 43'.6. Dec. 37°. H. A. 3h 20m.7 =8th C. 76° 18' or 76°. 
Table 2 at 76°. Dist. 100 = Diff. Lat. 24.2. Dep. 97.0. Diff. Long. 125 

Lat. Left 38° 43' 36" N Long. Left 59° 11'W 

Diff. Lat. 24 12 N Diff. Long. 2 5 E 

Lat. In 39 17 48 N Long. In 57 06 W = Dep. for 9th G. 


9th Course 

Lat. Dep. 39° 17'.8 Long. Dep. 57° 06' W 

Long. Dest. 9 W 

Diff. Long. 48 06 =H. A. 3h 12m.3. 

Azimuth Lat. 39° 17'.8. Dec. 37°. H. A. 3h 12m.3 =9th G. 77° 52' or 78°. 

Note: As it will be noted that the Azimuth has now increased more 
than 1° 30' in a distance of 100 miles, we will reduce the distance to be 
run on this course to 80 miles. 

Table 2 at 78°. Dist. 80= Diff. Lat. 16.6 Dep. 78.3. Diff. Long. 101. 

Lat. Left 39° 17' 48" N Long. Left 57° 06' W 

Diff. Lat. 16 36 N Diff. Long. 1 41 E 

Lat. In 39 34 24 N Long. In 55 25 W=Dep. for 10th G. 


10th Course 

Lat. Dep. 39° 34'.4 Long. Dep. 55° 25' W 

Long. Dest. 9 W 


Diff. Long. 46 25=H. A. 3h 5m.7 












8 


PRACTICAL ADVANCED NAVIGATION 


Azimuth Lat. 39® 34'.4. Dec. 37°. H. A. 3h 5m.7 = 10th C. 78° 55' or 79°. 
Table 2 at 79°. Dist. 80 =Diff. Lat. 15.3 Dep. 78.5. Diff. Long. 102. 

Lat. Left 39° 34' 24" N Long. Left 55° 25' W 
Diff. Lat. 15 18 N Diff. Long. 1 42 E 


Lat, In 39 49 42 N Long. In 53 43 W=Dep. for 11th C. 


11th Course 

Lat. Dep. 39° 49'.7 Long. Dep. 53° 43' W 

Long. Dest. 9 W 


Diff. Long. 44 43 =H. A. 2h 58m.9 

Azimuth Lat. 39° 49'.7. Dec. 37°. H. A. 2h 58m.9 = 11th C. 80° 2' or 80°. 
Table 2 at 80°. Dist. 80=Diff. Lat. 13.9. Dep. 78.8. Diff. Long. 103. 

Lat. Left 39° 49' 42" N Long. Left 53° 43' W 
Diff. Lat. 13 54 N Diff. Long. 1 43 E 


Lat. In 40 03 36 N Long. In 52 00 W=Dep. for 12th 0. 


IMh Course 

Lat. Dep. 40° 03'.6 Long. Dep. 52° W 

Long. Dest. 9 W 

Diff. Long. 43 =H. A. 2h 52m 

Azimuth Lat. 40° 3'.6. Dec. 37°. H. A. 2h 52m = 12th C. 81° 9' or 81°. 
Table 2 at 81°. Dist. 80 = Diff. Lat. 12.5. Dep. 79. Diff. Long. 103. 

Lat. Left 40° 03' 36" N Long. Left 52° 00' W 

Diff. Lat. 12 30 N Diff. Long. 1 43 E 

Lat. In 40 16 06 N Long, In 50 17 W=Dep. for 13th 0. 


13th Course 

Lat. Dep. 40° 16'.1 Long. Dep. 50° 17' W 

Long. Dest. 9 W 


Diff. Long. 41 17 =H. A. 2h 45m.l 

Azimuth Lat. 40° 16'.1. Dec. 37°. H. A. 2h 45m.l =13th C. 82° 19' or 82°. 
Table 2 at 82°. Dist. 80= Diff. Lat. 11.1. Dep. 79.2. Diff. Long. 104. 

Lat. Left 40° 16' 06" N Long. Left 50° 17' W 
Diff. Lat. 11 06 N Diff. Long. 1 44 E 


Lat. In 40 27 12 N Long. In 48 33 W =Dep. for 14th G. 













PRACTICAL ADVANCED NAVIGATION 


9 


l/iih Course 

Lat. Dep. 40® 27'.2 Long. Dep. 48® 33' W 

Long. Dost. 9° W 

Diff. Long. 39 33 =H. A. 2h 38m.2 

Azimuth Lat. 40® 27'.2. Dec. 37®. H. A. 2h 38m.2 = 14th C. 83® 25'. 
Table 2 at 83° 25'. Dist. 80 =Diff. Lat. 9. Dep. 79.5 Diff. Long. 104. 

Lat. Left 40® 27' 12" N Long. Left 48° 33' W 

Diff. Lat. 9 N Diff. Long. 1 44 E 

Lat. In 40 36 12 N Long. In 46 49 W =Dep. for 15th G. 


15th Course 

Lat. Dep. 40® 36'.2 Long. Dep. 46® 49' W 

Long. Dest. 9 W 

Diff. Long. 37 49 =H. A. 2h 31m.3 

Azimuth Lat. 40® 36'.2. Dec. 37®. H. A. 2h 31m.3 = 15th C. 84® 34'. 
Table 2 at 84° 34'. Dist. 80= Diff. Lat. 7.7. Dep. 79.7. Diff. Long. 105. 

Lat. Left 40® 36' 12" N Long. Left 46° 49' W 

Diff. Lat. 7 42 N Diff. Long. 1 45 E 

Latin 40 43 54 N Long. In 45 04 W=Dep. for 16th G. 


16th Course 

Lat. Dep. 40® 43'.9 Long. Dep. 45® 04' W 

Long. Dest. 9 W 


Diff. Long. 36 04=H. A. 2h24m.2 

Azimuth Lat. 40® 43'.9. Dec. 37®. H. A. 2h 24m.2 = 16th C. 85® 49' or 86®. 
Table 2 at 86°. Dist. 80 = Diff. Lat. 5.6. Dep. 79.8. Diff. Long. 105. 

Lat. Left 40® 43' 54" N Long. Left 45® 04' W 
Diff. Lat. 5 36 N Diff. Long. 1 45 E 


Lat. In 40 49 30 N Long. In 43 19 W = Dep. for 17th G. 


17th Course 

Lat. Dep. 40® 49'.5 Long. Dep. 43® 19' W 

Long. Dest. 9 W 

Diff. Long. 34 19 =H. A. 2h 17m.3 

Azimuth Lat. 40® 49'.5. Dec. 37®. H. A. 2h 17m.3 = 17th C. 86® 57' or 87®. 
Table 2 at 87®. Dist. 80 = Diff. Lat. 4.2. Dep. 79.9. Diff. Long. 106. 












10 


PRACTICAL ADVANCED NAVIGATION 


Lat. Left 40° 49' 30" N Long. Left 43° 19' W 
Diff. Lat. 4 12 N Diff. Long. 1 46 E 


Lat. In 40 53 42 N Long. In 41 33 W = Dep. for 18th G. 


18th Course 

Lat. Dep. 40° 53'.7 Long. Dep. 41° 33' W 

Long. Dest. 9 W 


Diff. Long. 32 33 =H. A. 2h. lOm.2 

Azimuth Lat. 40° 53'.7. Dec. 37°. H. A. 2h. lOm.2 = 18th C. 88° 6' or 88°. 
Table 2 at 88°. Dist. 80 =Diff. Lat. 2.8. Dep. 80.0. Diff. Long. 106. 

Lat. Left 40° 53' 42" N Long. Left 41° 33' W 
Diff. Lat. 2 48 N Diff. Long. 1 46 E 


Lat. In 40 56 30 N Long. In 39 47 W= Dep. for 19th G. 


19th Course 

Lat. Dep. 40° 56'.5 Long. Dep. 39° 47' W 

Long. Dest. 9 W 


Diff. Long. 30 47 =H. A. 2h 3m.l 

Azimuth Lat. 40° 56'.5. Dec. 37°. H. A. 2h 3m.l =19th C. 89° 19' or 89°. 
Table 2 at 89°. Dist. 80 = Diff. Lat. 1. 4. Dep. 80.0. Diff. Long. 106. 

Lat. Left 40° 56' 30" N Long. Left 39° 47' W 
Diff. Lat. 1 24 N Diff. Long. 1 46 E 


Lat. In 40 57 54 N Long. In 38 01 W=Dep. for 20th G. 


20th Course 

Lat. Dep. 40° 57'.9 Long. Dep. 38° 01' W 

Long. Dest. 9 W 

Diff. Long. 29 01 =H. A. Ih56m.l 

Azimuth Lat. 40° 57'.9. Dec. 37°. H. A. Ih 56m.l =20th C. 90° 31'. 
Table 2 at 90° 30'. Dist. 80 = Diff. Lat. 1'. Dep. 80. Diff. Long. 106. 

Lat. Left 40° 57' 54" N Long. Left 38° 01' W 

Diff. Lat. 1 00 S Diff. Long. 1 46 E 

Lat. In 40 56 54 N Long. In 36 15 W =Dep. for 21st G. 


21st Course 

Lat. Dep. 40° 56'.9 Long. Dep. 36° 15' W 

Long. Dest. 9 W 


Diff. Long. 27 15 =H. A. Ih 49m 














PRACTICAL ADVANCED NAVIGATION 


11 


Azimuth Lat. 40° 56'.9. Dec. 37°. H. A. Ih 49m =21st C. 91° 41' or 92°. 
Table 2 at 92°. Dist. 80 =Diff. Lat. 2.8. Dep. 80.0. Diff. Long. 106. 

Lat. Left 40° 56' 54" N Long. Left 36° 15' W 
Diff. Lat. 2 48 S Diff. Long 1 46 E 


Lat. In 40 54 06 N Long. In 34 29 W=Dep. for 22d G. 


22d Course 

Lat. Dep. 40° 54'.1 Long. Dep. 34° 29' W 

Long. Dest. 9 W 

Diff. Long. 25 29 =H. A. Ih 41m.9 

Azimuth Lat. 40° 54'.1. Dec. 37°. H. A. Ih 41m.9 =22d C. 92° 52' or 93°. 
Table 2 at 93°. Dist. 80= Diff. Lat. 4.2. Dep. 79.9. Diff. Long. 106. 

Lat. Left 40° 54' 06" N Long. Left 34° 29' W 

Diff. Lat. 4 12 S Diff. Long. 1 46 E 

Latin 40 49 54 N Long. In 32 43 W=Dep. for 23d G. 


23d Course 

Lat. Dep. 40° 49'.9 Long. Dep. 32° 43' W 

Long. Dest. 9 W 

Diff. Long. 23 43 =H. A. Ih 34m.9 

Azimuth Lat. 40° 49'.9. Dec. 37°. H. A. Ih 34m.9 =23d C. 94° 4' or 94°. 
Table 2 at 94°. Dist. 80= Diff. Lat. 5.6. Dep. 79.8. Diff. Long. 105. 

Lat. Left 40° 49' 54" N Long. Left 32° 43' W 

Diff. Lat. 5 36 S Diff. Long. 1 45 E 

Latin 40 44 18 N Long. In 30 58 W=Dep. for 24th G. 


24 th Course 

Lat. Dep. 40° 44'.3 Long. Dep. 30° 58' W 

Long. Dest. 9 W 

Diff. Long. 21 58 =H. A. Ih 27m.9 

Azimuth Lat. 40° 44'.3. Dec. 37°. H. A. Ih 27m.9 =24th C. 95° 16' or 95°. 
Table 2 at 95°. Dist. 80 =Diff. Lat. 7. Dep. 79.7. Diff. Long. 105. 

Lat. Left 40° 44' 18" N Long. Left 30° 58' W 

Diff. Lat. 7 00 S Diff. Long. 1 45 E 

Latin 40 37 18 N Long. In 29 13 W=Dep. for 25th G. 













12 


PRACTICAL ADVANCED NAVIGATION 


25th Course 

Lat. Dep. 40° 37'.3 Long. Dep. 29° 13' W 

Long. Dost. 9 W 

Diff. Long. 20 13 =H. A. Ih 20m.9 

Azimuth Lat. 40° 37'.3. Dec. 37°. H. A. Ih 20m.9 =25th C. 96° 35'. 
Table 2 at 96° 35'. Dist. 80 =Diff. Lat. 9'. Dep. 79.5. Diff. Long. 104. 

Lat. Left 40° 37' 18" N Long. Left 29° 13' W 

Diff. Lat. 9 00 S Diff. Long. 1 44 E 

Lat. In 40 28 18 N Long. In 27 29 W=Dep, for 26th G. 


26th Course 

Lat. Dep. 40° 28'.3 Long. Dep. 27° 29' W 

Long. Dest. 9 W 

Diff. Long. 18 29 =H. A. Ih 13m.9 

Azimuth Lat. 40° 28'.3. Dec. 37°. H. A. Ih 13m.9 =26th C. 97° 39'. 
Table 2 at 97° 39'. Dist. 80= Diff. Lat. 10.5. Dep. 79.3. Diff. Long. 104. 

Lat. Left 40° 28' 18" N Long. Left 27° 29' W 

Diff. Lat. 10 30 S Diff. Long. 1 44 E 

Lat. In 40 17 48 N Long. In 25 45 W=Dep, for 27th G. 


27th Course 

Lat. Dep. 40° 17'.8 Long. Dep. 25° 45' W 

Long. Dest. 9 W 

Diff. Long. 16 45 =H. A. Ih 7m. 

Azimuth Lat. 40° 17'.8. Dec. 37°. H. A. Ih 7m =27th C. 98° 51' or 99°. 
Table 2 at 99°. Dist. 80 = Diff. Lat. 12.5. Dep. 79. Diff. Long. 103. 

Lat. Left 40° 17' 48" N Long. Left 25° 45' W 

Diff. Lat. 12 30 S Diff. Long. 1 43 E 

Lat, In 40 05 18 N Long. In 24 02 W=Dep. for 28th G. 


28th Course 

Lat. Dep 40° 5'.3 Long. Dep. 24° 02' W 

Long. Dest. 9 W 

Diff. Long. 15 02=H. A. IhOm.l 

Azimuth Lat. 40° 5'.3. Dec. 37°. H. A. Ih Om.l =28th C. 99° 56' or 100°. 
Table 2 at 100°. Dist. 80 = Diff. Lat. 13.9. Dep. 78.8. Diff. Long. 103. 












PRACTICAL ADVANCED NAVIGATION 


13 


Lat. Left 40° 05' 18" N Long. Left 24° 02' W 
Diff. Lat. 13 54 S Diff. Long. 1 43 E 


Lat. In 39 51 24 N Long. In 22 19 W=Dep. for 29th C. 


S9th Course 

Lat. Dep 39° 51'.4 Long. Dep. 22° 19' W 

Long. Dest. 9 W 

Diff. Long. 13 19=H. A. Oh 53m.3 

Azimuth Lat. 39° 51.4. Dec. 37°. H.A. 53m.3 =29th C. 101° lO'or 101°. 
Table 2 at 101°. Dist. 80 =Di£f. Lat. 15.3. Dep. 78.5. Diff. Long. 103. 

Lat. Left 39° 51' 24" N Long. Left 22° 19' W 

Diff. Lat. 15 18 S Diff. Long. 1 43 E 

Lat. In 39 36 06 N Long. In 20 36 W=Dep. for 30th G. 


30th Course 

Lat. Dep. 39° 36'.1 Long. Dep. 20° 36' W 

Long. Dest. 9 W 


Diff. Long. 11 36=H. A. 0h46m.4 

Azimuth Lat. 39° 36'.1. Dec. 37°. H. A. 46m.4 =30th C. 102° 24'. 

Table 2 at 102° 24'. Dist. 80= Diff. Lat. 17.2. Dep. 78.1. Diff. Long. 102. 

Lat. Left 39° 36' 06" N Long. Left 20° 36' W 
Diff. Lat. 17 12 S Diff. Long. 1 42 E 


Lat. In 39 18 54 N Long. In 18 54 W=Dep. for 31st G. 


31st Course 

Lat. Dep. 39° 18'.9 Long. Dep. 18° 54' W 

Long. Dest. 9 W 


Diff. Long. 9 54 =H. A. Oh 39m.6 

Azimuth Lat. 39° 18'.9. Dec. 37°. H. A. 39m.6 =31st C. 103° 26'. 
Table 2 at 103° 26'. Dist. 80= Diff. Lat. 18.7. Dep. 77.7. Diff. Long. 100. 

Lat. Left 39° 18' 54" N Long. Left 18° 54' W 
Diff. Lat. 18 42 S Diff. Long. 1 40 E 


Lat. In 39 00 12 N Long. In 17 14 W =Dep. for 32d G. 


32d Course 

Lat. Dep. 39° 0'.2 Long. Dep. 17° 14' W 

Long. Dest. 9 W 


Diff. Long. 8 14 =H. A. 32m.9 














14 


PRACTICAL ADVANCED NAVIGATION 


Azimuth Lat. 39° 0'.2. Dec. 37°. H. A. 32m.9 =32d 0. 104° 50' or 105 
Table 2 at 105°. Dist. 80 = Dili. Lat. 20.7. Dep. 77.3. Diff. Long. 100. 

Lat. Left 39° 00' 12" N Long. Left 17° 14' W 
Diff. Lat. 20 42 S Diff. Long. 1 40 E 


Lat. In 38 39 30 N Long. In 15 34 W = Dep. for 33d O. 


SSd Course 


Lat. Dep. 38° 39'.5 Long. Dep. 15° 34' W 

Long. Dest. 9 W 


Diff. Long. 6 34=H.A. 26m.3 

Azimuth Lat. 38° 39'.5. Dec. 37°. H. A. 26m.3 =33d C. 106°. 

Table 2 at 106°. Dist. 80=Diff. Lat. 22.1. Dep. 76.9. Diff. Long. 98. 

Lat. Left 38° 39' 30" N Long. Left 15° 34' W 
Diff. Lat. 22 06 S Diff. Long. 1 38 E 


Lat. In 38 17 24 N Long. In 13 56 W=Dep. for 34th G. 


$4th Course 

Lat. Dep. 38° 17'.4 Long. Dep. 13° 56' W 

Long. Dest. 9 W 


Diff. Long. 4 56=H.A. 19m.7 

Azimuth Lat. 38° 17'.4. Dec. 37°. H. A. 19m.7 =34th C. 106° 45' or 107' 
Table 2 at 107°. Dist. 80 = Diff. Lat. 24.7. Dep. 76.1. Diff. Long. 98. 

Lat. Left 38° 17' 24" N Long. Lett 13° 56' W 
D. Lat. 24 42 S Diff. Long. 1 38 E 


Lat. In 37 42 42 N Long. In 12 18 W=Dep. for 35th G. 


S5th Course 

Lat. Dep. 37° 42'.7 Long. Dep. 12° 18' W 

Long. Dest. 9 W 


Diff. Long. 3 18=H. A. 13m.2 

Azimuth Lat. 37° 42'.7. Dec. 37°. H. A. 13m.2=35th G. 105° 40'or 106' 
Table 2 at 106°. Dist. 80=Diff. Lat. 22.1. Dep. 76.9. Diff. Long. 97. 

Lat. Left 37° 42' 42" N Long. Left 12° 18' W 
Diff. Lat. 22 06 S Diff. Long. 1 37 E 


Lat. In. 37 20 36 N Long. In 10 41 W=Dep. for 36th G. 













PRACTICAL ADVANCED NAVIGATION 


15 


36th or Final Course 

Lat. Dep. 37® 20'.6 Long. Dep. 10° 41' W 

Long. Dost. 9 W 


Diff. Long. 1 41 =H. A. 6m.7 

Azimuth Lat. 37° 20'.6. Dec. 37°. H. A. 6m.7 =36th C. 105° 48' or 106°. 
Table 2 at 106°. Dist. 80=Diff. Lat. 22.1. Dep. 76.9. Diff. Long. 96. 

Lat. Left 37° 20' 36" N Long. Left 10° 41' W 
Diff. Lat. 22 06 S Diff. Long. 1 36 E 


Lat. In 36 58 30 N Long. In 9 05 W 
Distance run on great circle courses by log 3040 miles. 

Note. —When the H. A. becomes less than 1 hour, to avoid such in¬ 
tricate interpolations, it is more practical to extend the Great Circle 
line beyond the point of destination. 

For example: On the Great Circle Course from Hatteras to Cape St. 
Vincent the destination may be extended to Lat. 25° 20' N. Long. 19° E. 


TO FIND GREAT CIRCLE DISTANCE 

Rule: 

With the Latitude of Departure as the Latitude. 

With the Latitude of Destination as the DecHnation. 

With the Difference of Longitude between Departure 
and Destination, converted into time, as the H. A. 

Take out the following Logs: 

Log Hav. of H. A. 

Log Cosine of Lat. 

Log Cosine of Deck 

Add these three Logs together. 

Opposite the Log Haversine of this Sum, read the Nat¬ 
ural Haversine. 

When the Latitude and Declination are same names, 
subtract the two. 

When the Latitude and Declination are different names, 
add the two. 

Take out the Natural Haversine of this Sum or Differ¬ 
ence, and add it to the Natural Haversine already 
obtained. 

The Natural Haversine of this Sum will equal the Zenith 
Distance. 

Reduce the Zenith Distance to Minutes of Arc and the 
result will be the Distance in Nautical miles on the 
Great Circle. 





16 


PRACTICAL ADVANCED NAVIGATION 


For Example: 


Lat. Dep. 35° 00' N 
Lat. Dest. 36 58 N 


Long. Dep. 73° 00' W 
Long. Dest. 9 05 W 


Diff.Long. 63 55 =H. A. 4h 15in 40s 


Log Haversine 4h 15m 
Log Cosine Lat. 35° 

Log Cos. Dec. 36° 58' 


40s =9.44741 


=9.91336 

=9.90254 


Lat. 35° 00' N 
Dec. 36 58 N 


Diff. 1 58 


Log Hav. 

Diff. 1° 58' 


9.26331= Nat. Hav. .18337 
Nat. Hav. .00029 


Nat. Hav. .18366 = 

Z.D.50°45'X60=3045 Dist. 


TO FIND INITIAL COURSE AND GREAT CIRCLE DISTANCE BY 
LOGARITHMS 

By this method we obtain the initial course and also the 
distance on a great circle between any two points. 


Rule: 


1. Put down the Lat. and Long, of Departure, and under 

them the Lat. and Long, of Destination. 

2. Find the Diff. of Long, between Departure and 

Destination. 

3. Take out the following Logs: 

Log Cosine of Difference of Long. 

Log Cotangent of Lat. of Destination. 

Add these two Logs together. 

4. The Log Tangent of this sum will equal the Auxihary 

Angle (0). 

5. When the Lat. of Departure and Destination are 

same names, add to the Auxiliary Angle (</>) the 
Lat. of Departure. 

WLen of contrary names subtract the two. 

6. Add together: 

Log Cosecant of Auxiliary Angle ( 0 ). 

Log Cosine of sum or difference of Auxiliary Angle 
and Lat. of Departure. 

Log Co-Tangent of Diff. of Long. 

7. The Log Co-Tangent of the Sum of these three logs 

will be the initial or first (true) Course. 






PRACTICAL ADVANCED NAVIGATION 


17 


TO FIND THE DISTANCE ON GREAT CIRCLE COURSES 
Rule: 

1. Add together: 

Log Tangent of the Sum or Difference of Auxihary 
Angle ( 0 ) and Lat. of Departure. 

Log Cosine of initial course. 

2. Log Co-Tangent of the sum of these two logs will be 

the Angular Distance. 

3. Reduce Angular Distance to Minutes, and the result 

will be the Distance on the great circle. 

For Example: 

Using same example as in Azimuth Method. 

Required the Great Circle Distance and Initial Course 
from Lat. 35° N, Long. 73° W to Lat. 36° 58' N, Long. 
9° 5' W? 

Lat. Dep. 35° 00 N Long. Dep. 73° 00' W 

Lat. Dest. 36 58 N Long. Dost. 9 05 W 

Diff. Long. 63 55 

Cos. Diff. Long. 63° 55' =9.64313 
Co-Tan. Lat. Des. 36 58 =0.12341 


Tangent 9.76654=0 Aux. Ang. 30° 17' 30" (Lat's 
Lat. Dep +35 same 

- name) 

Sum 65 17 30 

Log. Cosec. 0 (Aux. Angle) 30° 17' 30" =0.29722 
Log. Cosine Sum 65 17 30 =9.62118 

Log. Co-Tan. Diff. Long. 63 55 =9.68978 

Co-Tangent 9.60818 =Init. G. N. 67° 55' E 

Log. Tan. of Sum 65° 17' 30" =0.33713 
Log. Cos. of Course 67 55 30 =9.57498 

Co-Tangent 9.91211=50° 45' 30"X60 =3045.5 

Distance 

The Courses can be determined by this method the same 
as in previous Azimuth method, by allowing for distance 
run on each course and using new departure for next course. 

It must be remembered that all courses by these methods 
are true courses, and in order to make them it is necessary 
to allow in the proper direction the deviation and variation 
of compass. 







18 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 

Required Great Circle Courses and Distance by Azimuth 
Method and Great Circle Courses and Great Circle Dis¬ 
tance by Computation of Logs from the English Channel 
in Lat. 49° 50' N. Long. 6° 27' W. to off Abaco L. H. 
(Hole in the Wall) in Lat. 25° 50' N. Long. 77° W. 


Azimuth Method 

Lat. Dep. 49° 50' N Long. Dep. 6° 27' W 
Long. Dest. 77 W 


Di£f. Long. 70 33 =H. A. 4h 42m.2 

Az. Lat. 49° 50'. Dec. 25° 50'. H. A. 4h 42m.2=Init. or 1st C. 86° 30 

or N 86° 30' W 
or 273° 30' 

Table 2 at 273° 30'. Dist. 100 =Diff. Lat. 6.1. Dep. 99.8. Diff. Long. 130. 

Lat. Left 49° 50' 00" N Long. Left 6° 27' W 
Diff. Lat. 6 06 N Difif. Long. 2 10 W 


Lat. In 49 56 06 N Long. In 8 37 W= Dep. for 2d G. 


2 d Course 

Lat. Dep. 49° 56'.1 Long. Dep. 8° 37' W 

Long. Dest. 77 W 


Diff. Long. 68 23 =H. A. 4h 33m.5 
Az. Lat. 49° 56'.1. Dec. 25° 50'. H. A. 4h 33m.5 =2d G. N. 88° 12' W or 272°. 
This same rule will apply to Destination. 


To Find Great Circle Distance 


Log. Hav. H. A. 4h 42m 12s =9.52311 
Log. Cosine Lat. 49° 50' =9.80957 

Log. Cosine Dec. 25 50 =9.95427 


Lat. 49° 50'N 
Decl.25 50 N 


Diff. 24 00 


Log. Hav. 

Diff. 24° 


9.28695= Nat. Hav. .19362 
Nat. Hav. .04323 


58° 14' 45"X60=Dist. 34941 miles. 


Nat. Hav. .23685 = 

Z.D. 58° 14'45'! 


By Computation of Logs 

Lat. Dep. 49° 50' N Long. Dep. 6° 27' W 

Lat. Dest. 25 50 N Long. Dest. 77 W 


Diff. Long. 70 33 











PRACTICAL ADVANCED NAVIGATION 


19 


Log. Cos. Diff. Long. 70° 33'=9.52242 
Log. Cot. Lat. of Dest. 25 50 = .31503 


Tangent 9.83745 =<^ (Aux. Ang.) 34° 31' 10" 

Lat. Dep.+49 50 00 


Sum 84 21 10 

Log. Cosec. 0 34° 31' 10"= .24672 

Log. Cos. of Sum 84 21 10 = 8.99300 
Log. Cot. Diff. Long. 70 33 00 =9.54794 


Cotang. 8.78766 =Init. C. N 86° 29' 26" W 

Table 2 at N. 86° 30' W. Dis. 100 =D. Lat. 6.1. Dep. 99.8. D. Long. 130 

Lat. Left 49° 50' 00" N Long. Left 6° 27' W 
Diff. Lat. 6 06 N Diff. Long. 2 10 W 


Lat. In 49 56 06 N Long. In 8 37 W =Dep. for 2d G. 


2 d Course 

Lat. Dep. 49° 56' 06" Long. Dep. 8° 37' W 

Long. Des. 77 W 


Diff. Long. 68 23 

Log. Cos. Diff. Long. 68° 23'=9.56631 
Log. Cot. Lat. Dest. 25 50 = .31503 


Tangent 9.88134 =<^ (Aux. Ang.) 37° 16' 

Lat. Dep. +49 56 


Sum 87 12 

Log. Cosec. <t> 37° 16'= .21787 

Log. Cos. of Sum 87 12 = 8.68886 
Log. Cot. Diff. Long. 68 23 =9.59799 


Cotang. 8.50472 = 2d G. N. 88° 10' W. or 272° 

This same rule will apply to Destination. 


To Find the Great Circle Distance 

Long. Tan. 0+Lat. Dep. 84° 21' 10" =1.00488 
Log. Cos. of Init. Course 86 29 26" =8.78685 


Cotang. 9.79173 = 58° 14' 30" X 60 = 

Dist. 3494.5 

Great Circle Distance 3494.5 miles. By Sailing on Mercator Chart the 
Distance be 3627 miles. 












20 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 3 
(From Bowditch) 

Find the Great Circle Distance and Initial Course from Lat. 40° N, 
Long. 70° W to Lat. 30° S, Long. 10° W. 

Lat. Dep. 40° N Long. Dep. 70° W 

Lat. Dest. 30° S Long. Dest. 10° W 


Diff. Long. 60° 

Log. Cos. D. of Long. 60° =9.69897 
Log. Cot. Lat. Dest. 30 = .23856 


Tangent 9.93753 =</> (Aux. Ang.) 40° 53'.5 Lat’s Diff. 

Lat. Dep. —40° name 


Diff. 0 53 .5 

Log. Cosec. of <f> 40° 53'.5 =0.18400 

Log. Cos. of Diff. 0° 53 .5 =9.99995 
Log. Cot. D. of Long. 60° =9.76144 


Gotang. 9.94539 = Init. C. S 48° 35'.5 E 

or 131° 24'.5 


To Find Great Circle Distance 

Log. Tan. of <f> -Lat. Dep. 0° 53'.5 =8.19212 
Log. Cos. of Init. Course 48° 35'.5 = 9.82048 


8.01260=89° 24'.5X60 = 

5364.5 Miles, Dist. 


Cotang. 








CHAPTER III 


DOUBLE CHRONOMETER METHOD FOR DETERMINING 
BY COMPUTATION THE POSITION OF VESSEL BY 
TWO OBSERVATIONS 

The position of the ship may be determined by this 
method both accurately and easily without the use of plot¬ 
ting charts or graphic methods. The results obtained will 
be the same as plotting Sumner Lines. It is especially use¬ 
ful in cloudy weather when it is doubtful that the Meridian 
Altitude will be available on that day. It is applicable not 
only to sun observations, which are naturally preferred, but 
also to observations of two stars, or other celestial bodies. 

Rule: 

1. Take an observation of sun or any other body, using 

D. R. Lat., and determine Long, by same. 

2. Take out True Bearing of body observed from Azi¬ 

muth Table. 

Note: If True Bearing exceeds 90° subtract it 
from 180° and name in adjacent quadrant. 

For Example: 

Azimuth N 109° E is 180°-109° =S 71° E. 

3. Enter Table 47 (Bowditch) with Lat. at top of page 

and Azimuth at side and take out the correspond¬ 
ing Long. Factor. 

Example : 

Lat. 37° 25'. Az. 71°=Long. Factor .44. This is 
Factor I. 

4. In Case of the Sun Observed: After the sun 

changes its bearing, two or more points, take 
another observation. To the Lat. and Long, of 
first observation, apply the differences made to 
second observation, by allowing for the course 
and distance run in the interval. IVith this new 
Lat. compute the Long, for second observation. 

21 


22 


PRACTICAL ADVANCED NAVIGATION 


Take out Azimuth and Long. Factor as before. 
This will be Factor II. 

5. In Case of Simultaneous Observations op Any 

Body: If it is possible to observe two stars whose 
bearings alternate 45° or more and are in adja¬ 
cent quadrants, this will give an excellent fix. 
Each star or other body observed should be worked 
with the same dead reckoning latitude to obtain 
the longitude. If the observations are not simul¬ 
taneous, the latitude and longitude of the fost 
observation must be brought forward to the time 
of second observation by allowing for the course 
and distance run in the interval. 

After working the two problems for Long., the 
Azimuth and Long. Factor must be taken out for 
each as explained before. 

6. If both observations were in the same or opposite 

quadrants, subtract ( —) the smallest from the 
greatest factor. 

Note: If both observations were S and E, or N and 
E they would be in the same quadrant. 

If one observation bore S and E and the other N and 
W they would be in opposite quadrants (N and W 
being opposite to S and E). 

7. If the observations are in adjacent quadrants, add 

(+) the two factors. 

Note: If one bore N and E and the other S and E, 
they would be in adjacent quadrants. 

8. Find the difference of Long, between the Long, of 

the First Observation brought forward to Second 
Observation, and the Long, of the Second Obser¬ 
vation. 

9. Divide the Diff. of Long, by the Sum or Diff. of the 

two factors. The result will be the error in the 
Lat. used. 

10. Multiply Factor I by the Lat. Error. The result 

will be the correction for Long, of First Observa¬ 
tion brought forward to Second Observation. 
Apply this correction by rule explained later. 

11. Multiply Factor II by the Lat. Error. The result 

will be correction for Long, of Second Observation. 

12. Apply the Lat. Error to the Lat. used at Second 

Observation, by the rule as explained later and the 
result will be the correct latitude. 


PRACTICAL ADVANCED NAVIGATION 


23 


If the observations, and calculations are correct the two 
Longitudes will agree, and consequently the position of the 
ship will be correct. 

HOW TO APPLY THE CORRECTIONS FOR LATITUDE AND 
LONGITUDE 

If, after bringing forward the Long, of first observation to 
the time of second observation, it is found there is a differ¬ 
ence of Long, between them, it is evident that there must be 
an error in the Lat. used. 

If the bodies observed are in the same or opposite quad¬ 
rants, the error of Long, caused by error in Lat. used; is, 
for both first and second observations in the same direction, 
i.e., both East or both West. 

To determine in which way to apply this correction, we 
find the direction in which the error is increasing. For 
instance, in Example 1 (which is in same quadrants) we 
find 3' difference of Long, between first and second observa¬ 
tion. First observation gives Long. 69° 27' W. Second ob¬ 
servation 69° 30' W. It is here evident that the second 
observation is further West than the first observation, and 
the error in Lat. we are using is increasing the Long, to the 
Westward. 

It is evident then from this, that the corrections for 
Long, must be applied to the East, and the corrections for 
Lat. to the North, as the further South the Lat., in this 
instance, the further West the Long. 

A simple method how to determine how to apply these 
corrections is as follows: 

Diff. between Longs., second observation further West. 

Azimuth of Body S yE (opposites) 

Correction to be applied to Long. W 

Eastward, a diagonal line drawn 
from E shows Lat. Correction to 
be applied N. 

If the bodies are in adjacent quadrants, the error in 
Long, produced by Error in Lat. used, is one East and one 
West. 

Taking Example 2 we find. 

Long. 1st obs. 56° 17' 00" W 

Long. 2d obs. 55° 55' 45" W = Diff. Long. 21' 15" (2d 
obs. East of 1st obs.) 


24 


PRACTICAL ADVANCED NAVIGATION 


It is seen here that the 2d obs. is East of 1st obs. There¬ 
fore the 1st obs. will be corrected to the East and the 2d 
obs. to the West. 

Az, 1st Ohs, S yE 

Long, to be corrected E, makes Lat. W (opposites) 
correction North. 

Az, 2d Ohs. S yW 

Long, to be corrected W, makes E (opposites) 

Lat. correction N. 


EXAMPLE 1 
(Sun in Same Quadrant) 

March 4, 1922. At 7.30 A. M. Obs. Alt. Sun’s L. L. 
13° 21' 45". Dip 36 ft. Chronometer read Oh 15m 25s 
which was slow 2m 8s. Lat. D. R. 37° 25' N. Long. D. R. 
68° 30' W. 

At 10 A. M. Obs. Alt. Sun’s L. L. 37° 23'. Chronometer 
read 2h 45m. Ship run in interval S 84° W (True) 12 knots 
per hour. 

Required true position at 2d Obs.? 


1st Observat'Hbn 


Chronometer 

Oh 

15m 

25s 

Decl. 4d Oh 

6° 

37'.6 

Slow 

+ 

2 

8 

Corr. — 


.3 

G. M. T. 4d 

0 

17 

33 

True Decl. 

6 

37 .3S 

Eq. Time 

- 

11 

59 

P. D. 

96 

37 .3 

G. A. T. 4d 

0 

5 

34 




Obs. Alt. 

13° 

21' 

45" 




Corr. Table 46 

+ 

6 

15 




True Alt. 

13 

28 

00 




Lat. D. R. 

37 

25 


Sec. .10005 



P. D. 

96 

37 

18 

Cosec. .00290 



Sum 

147 

30 

18 




Half Sum 

73 

45 

9 

Cos. 9.44683 



Rem. 

60 

17 

9 

Sin. 9.93877 




Log. Hav. 9.48855 = L. A. T. 3d 19h 30m 20s 
G.A.T. 4 0 5 34 


Long. T. 4 35 14 

1st Obs. Long. 68° 48' 30" W 
Azimuth N 109° E or S 71° E. Factor I (Table 47) = .44. 









PRACTICAL ADVANCED NAVIGATION 


25 


Position Brought Forward to 2d Observation 

True course S 84° W. Dist. 30 = D. Lat. 3' S. D. Long. 38' W. 

Lat. 1st Obs. 37° 25' N Long. 1st Obs. 68° 48' 30" W 

Diff. Lat. 3 S Diff. Long. 38 00 W 


Lat. brought forward 37 22 N Long, brought forward 69 26 30 W 


Observation 


Chronometer 

Slow 

G. M. T. 4d 
Eq. Time 

G. A. T. 4d 


2h 45m 00s 

+ 28 


Obs. Alt. 37° 23' 00" 
Corr. + 9 


47 

11 


8 

58 


2 35 10 


Decl. 4d 2h 
Corr. 

True Decl. 
P. D. 


True Alt. 37 32 00 

Lat. 37 22 Sec. .09976 

P. D. 96 34 54 Cosec. .00287 


6° 35'.7 

.8 


6 34 .9 S 
96 34.9 


Sum 171 28 54 

Half Sum 85 44 27 Cos. 8.87080 

Rem. 48 12 27 Sin 9.87248 


Log. Hav. 8.84591 =L. A. T. 3d 21h 57m 09s 
G.A.T. 4 2 35 10 


Long.T. 4 38 1 

2d Obs. Long. 69° 30' 15" W. 
Azimuth N 140° E or S 40° E. Factor II = 1.49. 

Long.lstObs.br'tf'rw’d69° 26' 30" W Factor I = .44 
Long. 2d Obs. 69 30 15 W Factor II =1.49 


Diff. Long. 
Diff. Long. 

Diff. Factor 


3 45 D. Fact. 1.05 (same quad) 

3'. 7 

-= 3'.5 Error in Lat. used. 

1 .05 


Factor I .44x3'.5 = 1'.54 correction to Long, of 1st Obs. 
Long. 1st Obs. 69° 26' 30" W Az. S /E 
Corr. 1 32 E N^ W 


69 24 58 W = 1st Obs. Long, corrected. 

Factor II. 1.49x3'.5 =5'.215 correction to Long, of 2d Obs. 

Long. 2d Obs. 69° 30' 15" W Az. S /E Long, correction E 
Corr. 5 13 E N^ W makes Lat. correc. N. 


69 25 02 W=2d Obs. Long, corrected. 














26 


PRACTICAL ADVANCED NAVIGATION 


Lat. Obs. 37° 22" 00" N Position of vessel at 2d Obs. 

Lat. Error 3 30 N Lat. 37° 25' 30" N 

Corr. Lat. 37° 25' 30" N Long. 69° 25' W 

See Marcq St. Hilaire Method, Problem No. 1. 
Simmer “Tangent’' Method, Problem No. 1. 


EXAMPLE 2 

(Sun in Adjacent Quadrants) 

Jan. 1, 1922. At 8.32 A. M. Obs. Alt. Sun’s L. L. 
12° 46'. Dip 40 ft. Chron. read 12h 33m 24s which was 
fast 33m 35s. D. R. Lat. 33° 13' N. Long. 54° 35' W. 

Same date at 3.50 P. M. Obs. Alt. Sun’s L. L. 12° 35' 15". 
Chron. 8h 5m 2s, fast 33m 37s. 

Ship ran in interval 258° (True) 92 miles. 

Required position at 2d Obs.? 


1st Observation 


Chron. 12h 33m 24s True Dec. 23° 2'.7 S 
Fast - 33 35 P. D. 113 2.7 


G. M. T.31d 23 59 49 
Eq. Time — 3 28 


G.A.T.Sld 23 56 21 

Obs. Alt. 12° 46' 00" 
Corr. 4- 5 40 


True Alt. 12 51 40 

Lat.D.R. 33 13 Sec. .07748 

P. D. 113 2 42 Cosec. .03612 


Sum 159 07 22 

Half Sum 79 33 41 Cos. 9.25811 

Rem. 66 42 1 Sin 9.96305 


Log. Hav.9.33476=L.A.T. 31d 20h 18m 21s 
G.A.T. 31 23 56 21 


Lon. T 

1st Obs. Long. 54° 30' W 
Azimuth N 129° E or S 51° E. Factor I = 1.00. 


3 38 00 









PRACTICAL ADVANCED NAVIGATION 


27 


Position Bronght Forward to M Ohs. 

True course 258®. Dist. 92 =Diff. Lat. 19' S. Diff. Long. 107 W. 
Lat. 1st Obs. 33° 13' N Long. 1st Obs. 54° 30' W 

D. Lat. 19 S Diff. Long. 1 47 W 


Lat. forwarded 32 54 N Long, forwarded 50 17 W 


2d Observation 

Chron. 8h 05m 02s True Decl. 23° I'.l S 


Fast — 

33 

37 

P. D. U3 

G.M.T. Id 7 

31 

25 


Eq. Time — 

3 

38 


G.A.T. Id 7 

27 

47 


Obs. Alt. 12° 

35' 

15" 


Corr. 

+ 

5 

40 


True Alt. 12 

40 

55 


Lat. 

32 

54 


Sec. .07592 

P. D. 

113 

1 

6 

Cosec. .03603 

Sum 

158 

36 

01 


Half Sum 79 

18 

0 

Cos. 9.26873 

Rem. 66 

37 

05 

Sin 9.96278 


Log. Hav. 9.34346 = L. A.T. Id 3h 44m 04s 
G.A.T. 1 7 27 47 


Lon. T 3 43 43 

2d Obs. Long. 55° 55' 45" W 
Azimuth N 129° W or S 51° W. Factor 11 = 1.00. 

Long. 1st Obs. forwarded 56° 17' 00" W Factor 1 = 1.00 
Long. 2d Obs. 55 55 45 W Factor II = 1.00 


Diff. Long. 21 15 SumFac. 2.00 (adj.quad) 

Diff. Long. 21'.25 

-= lO'.O Error in Latitude used. 

Sum of Fac. 2 .00 

Factor I 1.00 XlO'.O = 10'.6 Correction to Long. 1st Obs. 

Lon. 1st Obs. 56° 17' 00" W Az. S /E As Lon. Corr. is E., 

Lon. Corr. 10 36 E W Lat. Corr. is N 


56 06 24 W =lst Obs. Long. Corrected 

Factor II 1.00xl0'.6 =10'.6. Correction to Long. 2d Obs. 
Lon. 2d Obs. 55° 55' 45" W Az. S /W As Lon. Corr. is W 
Lon. Corr. 10 36 W E Lat. Corr. is N. 


56 6 21 W 2d Obs. Long. Corrected. 














28 PRACTICAL ADVANCED NAVIGATION 


Lat. Obs. 32° 54' 00" N Position of vessel at 2d Obs. 
Lat. Error 10 36 N Lat. 33° 04' 36" N 


Correct Lat. 33 04 36 N Long. 56 06 23 W 


EXAMPLE 3 

Star and Sun in Adjacent Quadrants 

Jan. 9, 1922. At 6.20 A. M. Obs. Alt. * Vega bearing 
East 24° 33' 20". Chron. read 12h 33ni 8s which was fast 
34m 55s. Lat. D. R. 23° 50' 30" N. Long. D. R. 86° 15' W. 
Dip 40 ft. 

Same date at 8.05 A. M. Obs. Alt. Sun's L. L. 13° 21' 37". 
Chron. read 2h 16m 8s, fast 34m 56s. 

Ship ran in interval between observations 241° (true) 


18 miles. 

Required 

position of ship at 2d Obs.? 





1st 

Observation {Vega, East) 


Chron. 


Oh 33m 08s 

* Decl. 

38° 42 

!'.7N 


Fast 


- 34 

55 


P. D. 

51 17 

.3 


G. M.T. 8d 

23 58 

13 






Sun’s R. A 


19 9 

1 






Corr. 


3 

56 






G. S. T. 

43 11 

10 






Obs. Alt. 

24 

° 33' 

20" 






Corr. 

- 

8 

20 






True Alt. 

24 

25 

00 



*R. A. 

18h 34m 

16s 

Lat. 

23 

50 

30 

Sec. 

.03874 

*H. A. 

5 7 

59 E or — 

P. D. 

51 

17 

18 

Cosec. .10770 










L. S. T. 

13 26 

17 

Sum 

99 

32 

48 



G. S. T. 

19 11 

10 

Half Sum 

49 

46 

24 

Cos. 

9.81011 

Lon. T. 

5 44 

53 

Rem. 

25 

21 

24 

Sin 

9.63170 

Long, 1st ob. 86° 13' 15" W 


Log. Hav. 9.58825 = H. A. 
Azimuth N 56° E. Factor I = .74. 


Position Brought Forward to 2d Observation 

True course 241°. Dist. 18 =Di£f. Lat. 8.5 S. Diff. Long. 17 W. 
Lat. 1st Obs. 23° 50' 30" N Long. 1st Obs. 86° 13' 15" W 

Diff. Lat. 8 30 S Diff. Long. 17 00 W 


Lat. forwarded 23 42 00 N Long, forw’ded 86 30 15 W 













PRACTICAL ADVANCED NAVIGATION 


29 


M Observation (Sun) 


Chron. 

2h 16m 

08s 




Fast 

— 

34 

56 


True Decl. 22° 9'.7 







P. D. 112 9 .7 


G. M. T.9d 

1 

41 

12 




Eq. Time 

- 

7 

6 




G. A. T. 9d 

1 34 

6 




Obs. Alt. 

13' 

’ 21' 

37" 




Corr. 

+ 

5 

50 




True Alt. 

13 

27 

27 




Lat. 

23 

42 


Sec. 

.03826 L.A.T. 8d 19h 47m 

48s 

P. D. 

112 

9 

42 

Cosec. .03334 G. A. T. 9d 1 34 

6 

Sum 

149 

19 

9 


Lon. T 5 46 

18 

Half Sum 

74 

39 

34 

Cos. 

9.42252 


Rem. 

61 

12 

7 

Sin 

9.94267 Lon. 2d Obs. 86° 34' 30" 

W 


Log. Hav. 9.43679 =H. A. 
Azimuth N 122° E or S 58° E. Factor II = .68 

Long. 1st Obs. forw’ded 86° 30' 15" W Factor I = .74 
Long. 2dObs. 86 34 30 W Factor 11= .68 


Diff. Long. 4 15 Sum. Fac. 1.42 (adjac. quad). 

Diff. Long. 4.25 

-=3' Error in Latitude used. 

Sum Factors 1.42 

Factor I .74x3'=2'.22 Correction to Long, of 1st Obs. 

Long. 1st Obs. 86° 30' 15" W Az. N\ E As Long. Cor. is W 
Long. Corr. 2 13 W S Lat. Cor. is N 


86 32 28 W = 1st Obs. Long, corrected. 

Factor II .68x3' =2'.04 Correction to Long. 2d Obs. 

Long. 2d Obs. 86° 34' 30" W Az. S /E As Long. Cor. is E 
Long. Corr. 2 2 E N''^ W Lat. Cor. is N 


86 32 28 W =2d Obs. Long, corrected. 

Lat. Obs. 23° 42' N Position of vessel at 2d Obs. 
Lat. Error 3 N Lat. 23° 45' N 

- Long. 86° 32' 28" W 

Correct Lat. 23 45 N 












30 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 4 

Simultaneous Observations Two Fixed Stars in Opposite Quadrants 

May 19, 1922. P. M. at ship. Lat. D. R. 37° 50' N. 
Long 74° W. Obs. Alt. * Capella bearing West 22° 8'. 
Dip 41 ft. Chronometer read 7h 46m 47s which was slow 
4h 59m 24s. 

Immediately after Obs. Alt. * Spica bearing East 
36° 19'. Chronometer read 7h 50m 40s. Required position 
of vessel? 


1st Observation {Capella, West) 


Chronometer 


7h 46m 

47s 





Slow 

+ 

4 59 

24 

* Decl. 45' 

’ 55 

'.2N 






P. D. 44 

4 

.8 


G. M. T. 19d 

12 46 

11 





Sun’s R. A. 


3 45 

29 





Corr. 


2 

6 





G. S. T. 

16 33 

46 





Obs. Alt. 

22' 

’ 08' 00' 

t 





Corr. 

— 

8 40 


*R. A. 

5h 

. 10m 

56s 





♦H. A. 

6 

27 

41W or + 

True Alt. 

21 

59 20 






Lat. 

37 

50 

Sec. 

.10248 L.S.T. 

11 

38 

37 

P. D. 

44 

4 48 

Cosec. .15761 G. S. T. 

16 

33 

46 

Sum 103 

54 08 


Lon. T 

4 

55 

09 

Half Sum 

51 

57 04 

Cos. 

9.78982 




Rem. 

29 

57 44 

Sin 

9.69847 Long, let ob. 

ys” 47' 15" w 



Log 

. Hav. 

9.74838-H. A 




Azimuth N 48° W. Factor I = 1.14. 





2d Observation {Spica, East) 


Chronometer 7h 50m 40s 
Slow +4 59 24 


G. M. T. 19d 12 50 04 * Decl. 10° 45'.5S 

Sun's R. A. 3 45 29 P. D. 100 45.5 

Corr. 2 06 


G. S. T. 


16 37 39 











PRACTICAL ADVANCED NAVIGATION 31 


Obs. Alt. 

36° 

19' 

00" 

* R. A. 13h 21m 

07s 

Corr. 

— 

7 

35 

H. A. 1 38 

47Eor - 

True Alt. 

36 

11 

25 

L.S.T. 11 42 

20 

Lat. 

37 

50 


Sec. .10248 G.S.T. 16 37 

39 

P. D. 

100 

45 

30 

Cosec. .00770 Lon. T. - 






4 55 

19 

Sum 

174 

46 

55 



Half Sum 

87 

23 

27 

Cos. 8.65823 


Rem. 

51 

12 

02 

Sin 9.89173 Long. 2d Ob. 73° 49 

1' 45" W 


Log. Hav. 8.66014 =H. A. 
Azimuth S 30° E Factor II =2.20 


Long. 1st Obs. 73° 47' 15"W Factor I =1.14 

Long. 2d Obs. 73 49 45 W Factor II =2.20 


Diff. Long. 2 30 Diff. Fac. 1.06 (opposite quad) 

Diff. Long. 2'.5 

-=2'.5 Error in Lat. used. 

Diff. Fac. 1.06 


Factor I 1.14x2'.5 =2'.85 Correction to Long. 1st Obs. 

Long. 1st Obs. 73° 47' 15" W Az. N\ W Az. Long. Cor. is E 

Long. Corr. 2 50 E S ^E Lat. Cor. is N 


73 44 25 W = Correct Long. 1st Obs. 

Factor II. 2.20x2'.5 =5'.5 Correction to I^ong. 2d Obs. 

Long. 2d Obs. 73° 49' 45" W Az. S /E Az. Long. Cor. is E 
Long. Corr. 5 30 E N'^ W Lat. Cor. is N 


73 44 15 W = Correct Long. 2d Obs. 

Lat. D. R. 37° 50' 00" N Position of vessel at 2d Obs. 
Lat. Error 2 30 N Lat. 37° 52' 30" N 


Corr. Lat. 37 52 30 N Long. 73 44 20 W 


See Marcq St. Hilaire Problem No. 2. 












32 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 6 

Simultaneous Observation Moon and Sun in Adjacent Quadrants 

March 9, 1922. At 4.30 P. M. Obs. Alt. Moon\s 
L. L. 32° 15' bearing East. Dip 40 ft. Chronometer read 
9h 10m. Fast 3m 40s. Lat. D. R. 37° 28' N. Long. 63° 
50' W. 

At 4.34 P. M. Obs. Alt. Sun's L. L. 12° 20'. Chron. 
9h 14m 

Required position of vessel at 2d Obs.? 


1st Observation {Moon, East) 


Chron. 

9h 10m 00s 


Fast 


3 

40 

Decl. 8h 15° 46'.8 N Diff. 120 





Corr. — 6.6 

G. M. T. 9d 

9 

6 

20 


Sun’s R. A. 

23 

5 

34 

Tr. Dec. 15 40 .2 N 

Corr. 


1 

30 

P. D. 74 19.8 

G. S. T. 

32 

13 

24 






Horiz. Par. 60' 

Obs. Alt. 

32° 

15' 

00" 


Corr. 

+ 

59 

36 


Table 49 





True Alt. 

33 

14 

36 


Lat. 

37 

28 


Sec. .10034 

P. D. 

74 

19 

48 

Cosec. .01645 

Sum 

145 

02 

24 


Half Sum 

72 

31 

12 

Cos. 9.47766 

Rem. 

39 

16 

36 

Sin 9.80145 


R. A. 8h 7h 54m 06s Diff. 297 
Corr. + 2 44 



Cor. R. A. 

7 56 

50 


Moon’s H. 

A. 3 59 

23 E or 


L. S. T. 

3 57 

27 


G. S. T. 

8 13 

24 

Log. Hav. 9.39590 

Lon. T. 

4 15 

57 

Long. 1st Obs. 

63° 59' 

15" W. 

Azimuth S 85° E. 

Factor I = .11. 













PRACTICAL ADVANCED NAVIGATION 


33 


2d Observation {Sun, West) 


Chron. 

9h 

14m 

00s 

Decl. 8h 4° 

33'.6 

Fast 

— 

3 

40 

Corr. — 

1.2 

G. M. T. 9d 

9 

10 

20 

True Decl. 4 

32 .4S 

Eq. T. 

— 

10 

43 

P.D. 94 

32 .4 

G. A. T. 9d 

8 

59 

37 



Obs. Alt. 

12' 

’ 20' 

00" 



Corr. 

+ 

5 

36 



True Alt. 

12 

25 

36 



Lat. 

37 

28 


Sec. .10034 


P. D. 

94 

32 

24 

Cosec. .00136 


Sum 

144 

26 

00 



Half Sum 

72 

13 

00 

Cos. 9.48490 


Rem. 

59 

47 

24 

Sin 9.93661 



Log. Hav. 9.52321 =L. A.T. 9d 4h 42m 14s 
G. A. T. 9d 8 59 37 


Long. T. 4 17 23 

Long. 2d Obs. 64° 20' 45" W 
Azimuth S 74° W. Factor II = .355. 

Long. 1st Obs. 63° 59' 15" W Factor I = .11 
Long. 2d Obs. 64 20 45 Factor II = .355 


Diff. Long. 21 30 Sum Fac. .465 (Adjacent quad) 

Diff. Long. 21'.5 

-=46' Error in Lat. used 

Sum Fac. .465 

Factor I .11X46' =5'.06. Correction to Long. 1st Obs. 

Long. 1st Obs. 63° 59' 15" W Az. S v E Correction to Long. W 

Long. Corr. 5 00 W N makes Lat. Correction S 


64 04 15 W = Correct. Long, for 1st Obs. 

Factor IT .355x46'= 16'.33. Correction to Long. 2d Obs. 

Lon. 2d Obs. 64° 20' 45" W Az. Sv W Correction to Long. E 
Long. Corr. 16 20 E N ^E makes Lat. Correction S 


64 04 25 W 

Lat. D. R. 37° 28' N Position of vessel at 2d obs. 

Lat. Error 46 S Lat. 36° 42' N 

- Long. 64° 04' 20" W 

Corr. Lat. 36 42 N 

See Marcq St. Hilaire Problem No. 3. 

Sumner Tangent Method Problem No. 2 













CHAPTER IV 


CONSTRUCTION OF A MERCATOR CHART 

A Mercator chart is constructed on the principle that 
the earth is a flat surface instead of a sphere, and is by far 
the most generally used for navigation purposes. 

At the equator a degree of longitude is equal to a degree 
of latitude, but in receding from the equator and approach¬ 
ing the pole, while the degrees of latitude remain always 
of the same length (save for a slight change due to the fact 
that the earth is not a perfect sphere), the degrees of longi¬ 
tude become less and less. 

Since, in the Mercator Projection, the degrees of longi¬ 
tude are made to appear everywhere of the same length, it 
becomes necessary, in order to preserve the proportion that 
exists at different parts of the earth’s surface between degrees 
of latitude and degrees of longitude, that the former be 
increased from their natural lengths, and such increase must 
become greater and greater the higher the latitude. 

The length of the meridian, as thus increased, between 
the equator and any given latitude, expressed in minutes 
at the equator as a unit, constitutes the number of meridional 
parts corresponding to that latitude. Table 3 (Bowditch) 
is the table of meridional parts. 

As the Mercator Projection is a distortion of the earth’s 
surface it is plainly to be seen that a straight line drawn 
between two places on a Mercator Chart is not the shortest 
distance between them. As already explained in Chapter II 
the shortest distance between any two points is a Great 
Circle, as this follows the true lines of the earth’s surface. 
On a Great Circle course the ship is always headed for her 
port of destination, whereas on a Mercator course she is 
never headed for it until she is close to her destination. 

If the chart for which a projection is to be made includes 
the equator, the values to be measured off are given directly 

34 


PRACTICAL ADVANCED NAVIGATION 


35 


by Table 3. If the equator does not come upon the chart 
then the parallels of latitude to be laid down should be 
referred to a principal parallel, preferably the lowest paral¬ 
lel to be drawn on the chart. The distance of any other 
parallel of latitude from the principal parallel is then the 
difference of the values for the two taken from Table 3. 

In adapting the scale to be used for the Projection, it is 
generally desirable to select a sheet of paper large enough 
to protect against the scale being so small as to make the 
chart unworkable. It is at times necessary, though, if the 
size of paper is limited, and the chart is needed to cover a 
large surface, that the scale be small. 

In the following illustration the length of space between 
lowest and highest parallel of latitude, approximately 5°, is 
18 inches. This gives a fairly good workable chart. 

We will now proceed with following example and explan¬ 
ation, illustration of which is given. 

Example: Let a Projection be required for a chart of 
8° extent in longitude, between the parallels of Lat. 40° 30' 
N, and 45° 25' N, and let the space allowable on the paper 
between these parallels measure 18 inches. 

1. Enter Table 3 (Bowditch) and take out the merid¬ 

ional difference of latitude or meridional parts) for 
40° 30', the lowest parallel of Lat., which we find 
is 2646.8. Take out same for highest parallel 45° 
25' =3048.7 The difference 3048.7-2646.8 = 401.9. 
This is the value of the Meridional Arc between 
these latitudes. 

2. The paper space allowable between lowest and high¬ 

est parallels being 18 inches we divide 18 inches 

by 401.9 or = 0.0448 inch, which will 

401.9 

be the scale of the chart. We will call same 
0.045 inch. Each 1' of Long, on this chart will 
therefore measure 0.045 inch. 

3. Draw in the center of the sheet a straight line, and 

assume it to be the middle meridian of the chart 
(A). 

4. At the lower border of the sheet draw a line perpen¬ 

dicular to the first line, and assume this line to be 
parallel of latitude 40° 30' (B). 

5. From the intersection of the lines lay off on the par¬ 

allel, on each side of the middle meridian, 4° of 



36 


PRACTICAL ADVANCED NAVIGATION 


Long., or distances equal to 0.045x60x4 = 10.8 
inches (C). 

6. From these points draw lines parallel to the middle 

meridian. These will be the Eastern and Western 
lines of the chart (D). 

7. Next we proceed to find the distance between Lat. 

40° 30' and 41° as follows: Enter Table 3 and we 
find Mer. parts for 40° 30' equal 2646.8 for 41® 
equal 2686.2. Thus, 2686.2—2646.8=39.4 which 
is value of meriodinal arc between Latitudes. 
This value 39.4x0.045 inch = 1.773 inches, dis¬ 
tance between 40° 30' and 41°. 

8. Measure this distance on the three meridians from 

the lowest parallel and draw a straight fine 
between them. This line will represent Lat. 41°. 

9. Following same rule we find distance between 40° 30' 

and 42°. M. D. L. 40° 30'=2646.8, for 42° = 
2766.0. Thus, 2766.0-2646.8 = 119.2x0.045 inch 
= 5.364 inches. Distance between 40° 30' and 42°. 

10. Measure this distance as before from lowest parallel 

and draw a fine representing 42°. 

11. The same rule applies until highest parallel is 

attained, and we find the following distances: 

40° 30'to 43° = 9.0135 inches. 

40° 30' to 44° = 12.7215 inches. 

40° 30' to 45° = 16.497 inches. 

40° 30' to 45° 25' = 18.08 inches. 

12. The line drawn to represent Lat. 45° 25' will be the 

Northern limit of the chart. 

13. On the lowest and highest parallel measure from 

the middle meridian on each side four distances 
each equal to 2.7 inches and draw fines parallel 
to middle meridian. These will be the degrees of 
longitude. 

14. Divide each de^ee of Lat. and Long, into 60 equal 

parts, and if it is wished the Compass Rose may 
be drawn. This will complete the construction. 
Note: It is necessary in constructing a chart that a 
great deal of care is taken that the fines drawn are 
exactly true. A good draughtsman’s outfit con¬ 
sisting of drawing board, ''T” square, 45° and 
60° angles, drawing instruments, and a three- (3) 









































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PRACTICAL ADVANCED NAVIGATION 


37 


cornered measure divided into lOths, 20ths, 40ths, 
50ths, and 60ths of an inch are advisable. With 
this equipment the problem is very simple, and 
should be practiced and understood by every deck 
officer. 

See illustration. 


CHAPTER V 


SUMNER LINES BY TANGENT METHOD 

This is a very simple method of determining ship’s 
position by Sumner Line. In the original Sumner Method 
two Latitudes were assxuned and a sight for Longitude was 
worked by each of them. A line connecting the two posi¬ 
tions found was the Line of Position, and the ship’s position 
was somewhere on this line. By the Tangent Method it is 
only necessary to work one sight for Longitude with the 
Latitude of D. R. After plotting this position on the chart, 
a hne drawn at right angles to Body’s True Bearing (which is 
obtained from Azimuth Table) will be the Line of Position. 
If the observation made, and calculations for same are cor¬ 
rect, the ship’s position will be somewhere on this line. 

In using the sun another observation worked in the same 
manner, after the sun has changed its bearing two or more 
points, will give another position line. Where this line 
crosses the first line, after the first line is brought forward 
for course and distance run in interval, will be the ship’s 
position at second observation. 

Rule: 

1. Take observation and determine Longitude by same, 

using D. R. Lat. 

2. Take out Body’s True Bearing from Azimuth Table. 

3. Plot position found on Chart. 

4. Draw a line through position at right angles to Body’s 

True Bearing. (90° from it.) This will be First 
Line of Position. 

5. If there is an interval between first and second ob¬ 

servations, allow for comse and distance run in 
interval, and draw a line parallel to first line of 
position. This will be First Line of Position 
projected for course and distance. 

6. Take another observation after body has changed its 

bearing two or more points, and plot second line as 
before. In working this observation it is neces- 
38 


PRACTICAL ADVANCED NAVIGATION 


39 


sary to use the Lat. D. R. brought forward from 
first observation. 

7. Where the second line crosses the projection of first 

line, will be the ship^s position at second observa¬ 
tion. 

8. If the observations were simultaneous use the same 

D. R. Lat. for each observation. After determin¬ 
ing Long, of same, plot the position lines as be¬ 
fore. Where they cross one another will be ship^s 
position. 


EXAMPLE 1 

Two Observations of the Sun 


March 4, 1922. At 7.30 A. M. Obs. Alt. Sun^s L. L. 
13° 21' 45". Dip 36 feet. Chron. read Oh 15m 25s which 
was slow 2m 8s. Lat. D. R. 37° 25' N. Long. 68° 30' W. 

At 10 A. M. Obs. Alt. Sun's L. L. 37° 23'. Chron. 
read 2h 45m. Ship run in interval S 84° W. (True) 12 
knots per hour. 

Required position at second observation? 


1st Observation 


Chron. Oh 15m 25s 

Slow +2 8 


Decl. 4d0h 6° 37'.6 


Corr. 


.3 


G.M.T.4d 0 17 33 

Eq. T. - 11 59 


True Decl. 6 37 .3 S 

P. D. 96 37 .3 


G.A.T.4d 0 5 34 


Obs. Alt. 13° 21' 45" 
Corr. + 6 15 


True Alt. 13 28 00 
Lat.D.R.37 25 Sec. .10005 
P. D. 96 37 18 Cosec. .00290 


Sum 147 30 18 

Half Sum 73 45 9 Cos 9.44683 

Rem. 60 17 9 Sin 9.93877 


Log. Hav. 9.48855 =L. A. T. 3d 19h 30m 20s 
G.A.T.4d 0 5 34 


Long. T. 4 35 14 

1st Obs. Long. 68° 48' 30" W 


Azimuth N 109° E. 









40 


PRACTICAL ADVANCED NAVIGATION 


Position Brought Forward to 2d Observation 

True course S 84° W Dist. 30 =Di£f. Lat. 3' S. Diff. Long. 38' W. 
Lat. 1st Obs. 37° 25' 00" N Long. 1st Obs. 68° 48' 30" W 
Diff. Lat. 3 00 S Diff. Long. 38 W 


Lat. brought forw’d 37 22 00 N Long, br’t forw'd 69 26 30 W 


2d Observation 


Chronometer 


2h 

45m 

00s Decl. 4d 2h 

6° 

35'.7 

Slow 



+ 

2 

8 Corr. 

— 

.8 

G. M. T. 

4d 


2 

47 

08 True Decl. 

6 

34 .9 S 

Eq. T. 



- 

11 

58 P. D. 

96 

34.9 

G. A. T. 4d 


2 

35 

10 



Obs. Alt. 

o 

CO 

23' 

00' 

t 




Corr. 

+ 

9 






True Alt. 

37 

32 

00 





Lat. 

37 

22 

00 

Sec. 

.09976 



P. D. 

96 

34 

54 

Cosec. .00287 



Sum 

171 

28 

54 





Half Sum 

85 

44 

27 

Cos. 

8.87080 



Rem. 

48 

12 

27 

Sin 

9.87248 




Log. Hav. 8.84591 =L. A. T. 3d 21h 57m 09s 
G.A.T. 4d 2 35 10 


Lon.T. 4 38 1 

2d Obs. Long. 69° 30' 15" W. 

Azimuth N 140° E. 

See: Chartlet following this problem. 

Double Chronometer Method, Ex. 1, 

Marcq St. Hilaire Method, Ex. 1. 











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PRACTICAL ADVANCED NAVIGATION 


41 


EXAMPLE 2 

Simultaneous Observations of Moon and Sun 

March 9, 1922. At 4.30 P. M. Obs. Alt. Moon’s L. L. 
32° 15' bearing East. Dip 40 feet. Chronometer read 
9h 10m which was fast 3m 40s. Lat. D. R. 37° 28' N. 
Long. 63° 50' W. 

At 4.34 P. M. Obs. Alt. Sun’s L. L. 12° 20'. Chronom¬ 
eter 9h 14m. Required position of vessel at 2d Obs.? 


1st Observation {Moon, East) 


Chronometer 

9h 10m 

00s 


Fast 

— 

3 

40 

Decl. 8h 15“ 46'.8 N Diff, 





Corr. — 6.6 

G. M. T. 9d 

9 

6 

20 


Sun’s R. A. 

23 

5 

34 

Tr. Decl. 15 40 .2 N 

Corr. 


1 

30 

P. D. 74 19 .8 

G. S. T. 

32 

13 

24 






Horizontal Parallax 60' 

Obs. Alt. 

CO 

to 

o 

15' 

00" 


Cor. Table 49 + 

59 

36 


True Alt. 

33 

14 

36 


Lat. 

37 

28 


Sec. .10034 

P. D. 

74 

19 

48 

Cosec. .01645 

Sum 

145 

02 

24 


Half Sum 

72 

31 

12 

Cos. 9.47766 

Rem. 

39 

16 

36 

Sin 9.80145 


R. A. 8h 

7h 54m 

06s (Diff. 297) 

Corr. 

+ 

2 

44 

Cor. R.A. 

7 

56 

50 

Moon’s H. A. 

3 

59 

23 E or — 

L. S. T. 

3 

57 

27 

G. S. T. 

8 

13 

24 


Log. Hav. 9.39590 = H.A. Lon. T 4 15 57 

Azimuth S 85° E or N 95° E. Long. 1st Obs. 63° 59' 15" W'. 

M Observation {Sun, West) 

Chronometer 9h 14m 00s Decl. 8h 4° 33'.6 

Fast - 3 40 Corr. - 1 .2 


G.M.T. 9d 9 10 20 
Eq. T. - 10 43 


G.A.T.Od 


8 59 37 


True Decl. 4 32 .4 S 
P. D. 94 32 .4 














42 


PRACTICAL ADVANCED NAVIGATION 


Obs. Alt. 12® 20' 00" 
Corr. + 5 36 


True Alt. 12 25 36 

Lat. 37 28 Sec. .10034 

P. D. 94 32 24 Cosec. .00136 


Sum 144 26 00 

Half Sum 72 13 00 Cos. 9.48490 

Rem. 59 47 24 Sin 9.93661 


Log. Hav. 9.52321 =L. A. T.9d 4h 42m 14s 
=G.A.T.9d 8 59 37 


Long. T. 4 17 23 

Azimuth S 74® W or N 106® W. Long. 2d Obs. 64® 20' 45" W. 

See: Chartlet following this problem. 

Double Chronometer Method, Ex. 5. 

Marcq St. Hilaire Method, Ex. 3. 






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CHAPTER VI 


MARCQ ST. HILAIRE METHOD 

This method of plotting position hnes is now being used 
extensively in the Merchant Service and the Navy. 

There are three formulas which may be used, namely, the 
Cosine-Haversine Formula, the Haversine Formula, and the 
Sine-Cosine Formula. The rules for each are here given. 

The position of the ship by dead reckoning must always 
be used to work from, and in the case where a course and 
distance is given in the interval between observations, the 
position by D. R. must be found for second observation 
before working problem. 


RULE FOR COSINE-HAVERSINE FORMULA 

In Case of the Sun Observed: 

1. Apply to G. M. T. the Equation of Time and find 

G. A. T. 

2. Reduce the Long. D. R. to Time, and apply to G. A. T. 

If West Long., subtract, if East, add. 

Result will be L. A. T. or Hour Angle (H. A.) 

In Case of Moon, Star or Planet Observed : 

1. Apply Long, in Time to G. M. T. as before. 

Result will be Local Mean Time (L. M. T.). 

2. Add to L. M. T. the Sun^s Right Ascension (S. R. A.) 

from page 2 (Nautical Almanac) and Correction 
from Table below for G. M. T. 

Result will be Local Siderial Time (L. S. T.) 

3. From L. S. T. subtract Right Ascension of Body 

Observed. 

Result will be Hour Angle of Body. 

43 


44 


PRACTICAL ADVANCED NAVIGATION 


In All Cases: 

1. Take out True Declination of Body Observed. 

2. Correct observed altitude and obtain true altitude. 

3. Take out the following Logs: 

From Table 45 (Bowditch) Log Haversine of Hour 
Angle. 

From Table 44 (Bowditch) Log Cosine of Latitude 

D. B. 

From Table 44 (Bowditch) Log Cosine of True 
Declination. 

Add these three logs together and subtract 20 from 
index number. 

4. Opposite the Log Haversine corresponding to sum 

of logs, read the Natural Haversine. 

5. If Latitude and Declination are same name, subtract 

less from greater. 

If Latitude and Declination are contrary name, add 
the two. 

6. Take out the Natural Haversine of Sum or Differ¬ 

ence of Latitude and Declination, and add to it 
the Natural Haversine obtained before. 

7. Nat. Hav. corresponding to the Sum of the two will 

be the Zenith Distance (Z. D.). 

8. Subtract Z. D. from 90°. Result will be Computed 

Altitude. 

9. Take out True Bearing of Body from Azimuth Table. 

10. Find the difference between the Computed Altitude 

and the True Altitude. Result will be the Alti¬ 
tude Difference or Intercept. 

11. If the True Altitude is greater than Computed Alti¬ 

tude, measure from the Dead Reckoning Position 
on the Line of Azimuth toward the Body a dis¬ 
tance equal to the Altitude Difference or Inter¬ 
cept, and draw the position line through this point 
at right angles to true bearing. If True Altitude 
is less than Computed Altitude, measure away 
from Body. 

In using the signs, + means toward the Body, — 
means away from Body, 


PRACTICAL ADVANCED NAVIGATION 


45 


RULE FOR HAVERSINE FORMULA 

1. Obtain True Declination, True Altitude and Hour 

Angle of Body as before. 

2. Find the Polar Distance. 

3. Subtract the Lat. D. R. from 90°. 

Result will be Co. Lat. 

4. Add together Co. Lat. and Polar Distance, and take 

out the Nat. Hav. of sum. 

5. Subtract the Co. Lat. from Polar Distance, and take 

out the Nat. Hav. of difference. 

6. Subtract the smallest from Greatest Nat. Hav. 

The result will be Nat. Hav. A. 

7. Take out the following logs: 

Log Haversine of Nat. Hav. A. =Log Haversine A. 
Log Haversine of Hour Angle. 

Add these two Logs together. Result will be Log 
Haversine B. 

8. Take out the following Logs: 

Natural Haversine of Log Haversine B. = Nat. 
Hav. B. 

Nat. Hav. of Co. Lat. — P. D. 

Add these two Logs together. 

9. Nat. Hav. corresponding to Sum will be Zenith Dis¬ 

tance. 

10. With the Zenith Distance find the Altitude Differ¬ 
ence as in previous formula. 

RULE FOR SINE-COSINE FORMULA 

1. Obtain True Declination, True Altitude and Hour 

Angle of Body as before. 

2. Reduce the Hour Angle to Arc. 

3. Take out the following Logs: 

Log Sine of Lat. 

Log Sine of Deck 

Add these two Logs. Result will be Log A. 

4. Take out the following logs: 

Log Cosine of Hour Angle. 

Log Cosine of Latitude. 

Log Cosine of Declination. 

Add these three Logs. Result will be Log B. 

5. Take out the following Logs: 

Nat. Hav. of Log Hav. A. 

Nat. Hav. of Log Hav. B. 


46 


PRACTICAL ADVANCED NAVIGATION 


6. Apply these logs, as follows: 

When Lat. and Decl. are same name, and Lat. is 
greater than Decl. 

Add together Nat. Hav. Log A and B. 

When Lat. and Decl. are same name, and Decl. is 
greater than Lat. Subtract less from greater. 

When Lat. and Decl. are different names, Subtract. 

7. Nat. Sine (Table 41) corresponding to sum or differ¬ 

ence will be the Computed Altitude. 

8. Find altitude difference as given in previous rule. 

EXAMPLE 1 

Two Observations of the Sun 

March 4, 1922. At 7.30 A. M. Obs. Alt. Sun^s L. L. 
13® 21' 45". Dip 36 feet. Chronometer read 12h 15m 25s 
which was slow 2m 8s. Lat. D. R. 37® 25" N. Long. 
68® 30' W. 

At 10 A. M. Obs. Alt. Sun's L. L. 37® 23'. Chron. read 
2h 45m. 

Ship course in interval was S 84® W. (True) 12 knots per 
hour. 

Required Intercept or Alt. Diff. by Cosine-Haversine 
formula, and also position of vessel by plotting position 
lines on Mercator Chart? (See Chart.) 

Gosine-Haversine Formula 


1st Observation 


Chron. 

Oh 15m 

25s 

True Decl. 

6° 

37' 

18" S 

Slow 

+ 2 

8 

Lat. D. R. 

37 

25 

00 N 

G.M.T.4d 0 17 

33 

Sum 

44 

02 

18 

Eq. T. 

- 11 

59 





G. A. T. 

0 5 

34 

Obs. Alt. 

130 21' 

45'J 

Long. W 

- 4 34m 

00s 

Gorr. 

+ 

6 

15 

L. A. T. 

19 31 

34 

True Alt. 

13 

28 

00 


Log. Hav. H. A. 19h 31m 34s-9.48502 

Log. Cosine Lat. 37° 25' -9.89995 

Log. Cosine Decl. 6° 37' 18" -9.99710 


Log. Hav. 9.38207-Nat.Hav. .24103 
Sum of Lat. and Decl. 44° 2' 18" - Nat. Hav. . 14056 


Nat. Hav. .38159-Z.D. 













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PRACTICAL ADVANCED NAVIGATION 


47 


Zen. Dist. 76® 18' 
90 


Comp. Alt. 13 42 True Azimuth N. 109® E 

True Alt. 13 28 


Alt. Diff. - 14 


Position Brought Forward to 2d Observation 

True Course S 84® W. Dist. 30 = Diff. Lat. 3' S. Diff. Long. 38' 
Lat. D. R. 1st Obs. 37® 25' N Long. D. R. 1st Obs. 68° 30' 

Diff. Lat. 3 S Diff. Long. 38 


Lat. D. R. 2d Obs. 37 22 N Long. D. R. 2d Obs. 69 08 W 


2d Observation 


Chronometer 

2h 45m 

00s 

True Decl. 

6® 

34' 

54" 

S 

Slow 

+ 2 

08 

Lat. D. R. 

37 

22 


N 

G. M. T. 

2 47 

08 

Sum 

43 

56 

54 


Eq. T. 

- 11 

58 






G. A. T. 

2 35 

10 

Obs. Alt. 

o 

CO 

23' 



Long. W. 

-4 36 

32 

Corr. 

+ 

9 



L. A. T. 

21 58 

38 

True Alt. 

37 

32 




Log. Hav. H. A. 21h 58m 38s=8.83560 
Log. Cosine Lat. 37® 22' = 9.90024 

Log. Cosine Decl. 6® 34' 54" =9.99713 


Log. Hav. 8.73297= Nat. Hav. .05407 
Sum of Lat. and Decl. 43® 56' 54"=Nat. Hav. .14002 


Nat. Hav. . 19409 =Z. D. 

Zen. Dist. 52® 16' 45" 

90 


Comp Alt. 37 43 15 Azimuth N 140® E 

True Alt. 37 32 


Alt. Diff. - 11 15 

See: Chartlet following this problem. 

Double Chronometer Method, Ex. 1. 
Sumner Tangent Method, Ex. 1. 

















48 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 2 

Simultaneous Observations of Two Fixed Stars 

May 19, 1922. P. M. at ship. Lat. D. R. 37° 50' N. 
Long. D. R. 74° W. Obs. Alt. Star Capella bearing West 
22° 08'. Dip 41 feet. Chronometer read 7h 46m 47s which 
was slow 4h 59m 24s. 

Immediately afterwards Obs. Alt. *Spica bearing East 
36° 19'. Chronometer 7h 50m 40s. 

Required Alt. Diff. by Haversine Formula, and also 
position of vessel by plotting position lines on Mercator 
Chart. (See Chart.) 

Haversine Formula 
1st Observation—Capella in the West 
Chronometer 7h 46m 47s Decl. 45® 55' 12" N 


Slow 

+ 4 

59 

24 P. D. 

44 

04 

48 

G.M.T. 19d 

12 

46 

11 




Long. 

- 4 

56 





L. M. T. 

7 

50 

11 




S. R. A. 

3 

45 

29 Obs. Alt. 

22° 

08' 


Corr. 


2 

06 Corr. 

- 

8' 

40" 

L. S. T. 

11 

37 

46 True Alt. 

21 

59 

20 

*R. A. 

5 

10 

56 




*H.A. 

6 

26 

50 




90°-Lat. 37° 

50'= Co. Lat. 52° 10' 





Co. Lat. 52° 10'+P. D. 44° 4' 48" =96° 14' 48" Nat. Hav. = .55440 
Co. Lat. 52° 10'-P. D. 44° 4' 48"= 8° 5' 12" Nat. Hav. = .00497 


Nat. Hav. A . 54943 


Nat. Hav. A. .54943 = Log. Hav. A. 9.73992 
H. A. 6h 26m 50s = Log. Hav. 9.74695 


Log. Hav. B 9.48687 

Nat. Hav. of Log. Hav. B. 9.48687 = .30681 

Nat. Hav. Go. Lat. -P. D. 8° 15' 12" = . 00497 


Nat. Hav. .31178 =Z.D. 











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PRACTICAL ADVANCED NAVIGATION 


49 


Zen. Dist. 67° 53' 15" 
90 


Comp. Alt. 22 06 45 True Azimuth W 48° W 

True Alt. 21 59 20 


Alt. Diff. - 7 25 


2d Observation—Spica in the East 


1st Chron. Time 

7h 46m 

47s 

Decl. 10° 

' 45' 

2d Chron. Time 

7 50 

40 

P. D. 100 

45 

Diff. 

3 

53 



1st L. S. T. 

11 37 

46 

Obs. Alt. 

36° 




Corr. 

— 

2d L. S. T. 

11 41 

39 



♦R. A. 

13 21 

07 

True Alt. 

36 


*H. A. 

90°-Lat. 37° 50' 


1 39 28 

=Co. Lat. 52° 10' 


Co. Lat. 52° 10'+P. D. 100° 45' 30" = 152° 55' 30" Nat. Hav. = .94521 
Co. Lat. 52° 10'-P. D. 100° 45' 30"= 48° 35' 30" Nat. Hav. = .16929 


Nat. Hav. A .77592 


Nat. Hav. A. . 77592 = Log. Hav. A. 9.88981 
H. A. Ih 39m 28s = Log. Hav. 8.66610 


Log. Hav. B. 8.55591 

Nat. Hav. of Log. Hav. B. 8.55591 = . 03597 
Nat. Hav. Co. Lat. -P. D. 48° 35' 30" = . 16929 


Nat. Hav. .20526 =Z.D. 


Zen. Dist. 53° 52' 45" 
90 


Comp. Alt. 36 07 15 True Azimuth N 149° E 

True Alt. 36 11 25 


Alt. Diff. + 4 10 

See: Chartlet following this problem. 

Double Chronometer Method, Ex. 4. 













50 


PRACTICAL ADVANCED NAVIGATION 


EXAMPLE 3 

Simultaneous Observations of Moon and Sun 

March 9, 1922. At 4.30 P. M. Obs. Alt. Moon. L. L. 
32® 15' bearing East. Dip 40 feet. Chronometer read 
9h 10m which was fast 3m 40s. Lat. D. R. 37® 28' N. Long. 
63® 50' W. 

At 4.30 P. M. Obs. Alt. Sun's L. L. 12® 30'. Chronom¬ 
eter 9h 14m. Required Alt. Diff. by Sine-Cosine Formula, 
and position of vessel by plotting position lines on Mercator 
chart? 


Sine-Cosine Fobmula 


Ut Observation—Moon in the East 


Chronometer 

9h 

10m 

00s 

Decl. 8h 

15° 

46'.8 


Fast 

— 

3 

40 

Corr. 

- 

6.( 

5 


G. M. T. 9d 

9 

6 

20 

True Decl. 

15 

40 .1 

2N 


Long. - 

• 4 

15 

20 






L. M. T. 

4 

51 

00 

R. A. 8h 

7h , 

54m 

06s 


S. R. A. 

23 

5 

34 

Corr. 

+ 

2 

44 


Corr. 


1 

30 










Corr. R. A. 

7 

56 

50 


L. S. T. 

27 

58 

04 






Moon’s R. A. 

7 

56 

50 

Obs. Alt. 


32° 

15' 

00" 





Corr.Table 49 

+ 

59 

36 


20 

01 

14 







24 



True Alt. 


33 

14 

36 

H.A. 

3 

58 

46 =59° 

41' 30" 






Log. Sine Lat. 37° 28' =9.78412 

Log. Sine Decl. 15° 40' 12" =9.43152 


Log. A 9.21564 

Log. Cosine H. A. 59° 41' 30" =9.70299 
Log. Cosine Lat. 37° 28' = 9.89966 

Log. Cosine Decl. 15° 40' 12" =9.98355 


Log.B 9.58620 

Nat. Hav. Log. A 9.21564 = . 16430 
Nat. Hav. Log. B 9.58620 = .38565 


Nat. Sine (Table 41) .54995 = Comp. Alt. 33° 21' 45" 
True Alt. 33 14 36 


True Azimuth N 95° E 


Alt. Diff. 


7 09 



































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To face page 51 












































PRACTICAL ADVANCED NAVIGATION 


51 


2d Observation—Sun in the West 


Chronometer 9h 14m 

00s 

Decl. 

4° 32'.4 S 

Fast 

- 3 

40 



G. M. T. 

9 10 

20 

Obs. Alt. 

12° 20' 00" 

Eq. T. 

- 10 

43 

Corr. 

+ 5 36 

G. A. T. 

8 59 

37 

True Alt. 

12 25 36 

Long. 

- 4 15 

20 



L. A. T. 

4 44 

17=71° 

04'15" 



Log. Sine Lat. 37° 28' =9.78412 

Log. Sine Decl. 4° 32' 24" =8.89847 


Log.A 8.68259 

Log.CosineH.A. 71° 4' 15"=9.51108 
Log. Cosine Lat. 37° 28' =9.89966 

Log. Cosine Decl. 4° 32' 24" =9.99864 


Log. B 9.40938 

Nat. Hav. Log. A 8.68259 = .04815- 
Nat. Hav. Log. B 9.40938 = .25667 


Nat. Sine .20852= Comp. Alt. 12° 02' 15" 
True Alt. 12 25 36 


Azimuth N 106° W Alt. Diff. + 23 21 

See: Chartlet following this problem. 

Double Chronometer Method, Ex. 5. 
Sumner "Tangent" Method, Ex. 2. 










52 


PRACTICAL ADVANCED NAVIGATION 


EXTRACTS FROM THE AMERICAN NAUTICAL ALMANAC 
FOR THE YEAR 1922 

RIGHT ASCENSION OF THE MEAN SUN AT GREENWICH 
MEAN NOON 


JANUARY, 1922 AUGUST, 1922 


Day 

8 

19h 9m 

Os .9 

Day 

9 

9h 

8m 

47s.O 

9 

19 12 

57 .5 

10 

9 

12 

43 .6 

10 

19 16 

54 .0 

11 

9 

16 

40 .1 


MAY, 1922 


DECEMBER, 

1922 

Day 

18 

3h 41m 

32s.9 

Day 

8 

17h 

5m 

50s.0 

19 

3 45 

29 .4 

9 

17 

9 

46 .5 

20 

3 49 

26 .0 

10 

17 

13 

43 .1 



JUNE, 

1922 





Day 

4 4h 48m 34s.3 

5 4 52 30 .9 

6 4 56 27 .4 

THE SUN 
JANUARY, 1922 



Sunday 1 


- 7 -- 

Monday 9 


G. M. T. 

Sun’s 

Equation 

G. M. T. 

Sun’s 

Equation 

Hrs. 

Declination 

of Time 

Hrs. 

Declination 

of Time 

0 

-23° 2'.7 ■ 

- 3m 

28s.O 

0 

-22° 10'.4 - 

• 7m 3s.7 

2 

23 2 .3 

3 

30 .4 

2 

22 9.7 

7 5 .8 

4 

23 1.9 

3 

32 .8 

4 

22 9.0 

7 7 .8 

6 

23 1.5 

3 

35 .2 

H. D. 

0.4 

1 .0 

8 

23 1.1 

3 

37 .6 




10 

23 0.7 

3 

39 .9 


Semi-Diameter 


H. D. 

0.2 


1 .2 


Jan. 1st 16'.30 


11 16.29 


MARCH, 1922 

Satueday 4 Thuesday 9 


G. M. T. 

Sun’s 

Equation 

G. M. T. 

Sun’s 

Equation 

Hrs. 

Declination 

of Time 

Hrs. 

Declination 

of Time 

0 

- 6° 37'.6 - 

-11m 59s.6 

6 

- 4° 35'.5 

— 10m 45s.4 

2 

6 35.5 

11 58 .5 

8 

4° 33 .6 

10 44 .1 

4 

6 33.8 

11 57 .5 

10 

4 31.6 

10 42 .9 

H. D. 

1.0 

0 .5 

H. D. 

1 .0 

0 .6 


Semi-Diameter 
March 1 16'.17 

11 16.13 


PRACTICAL ADVANCED NAVIGATION 


53 


APPARENT PLACES OF STARS, 1922 
FOR THE UPPER TRANSIT OF GREENWICH 


Star 

Month 


Declination 


Right Ascension 

Vega 

Jan. 1st 


+38° 42'.7 


18h 34m 16S.4 

Dec. 1st 


38 43.0 


18 

34 

18 .2 

Capella 

May 1st 


+45 55.2 


5 

10 

55 .6 


Dec. 1st 


45 55.1 


5 

10 

61 .4 

Spica 

May 1st 


-10 45.5 


13 

21 

7.3 

Fomalhaut 

xAug. 1st 


-30 1 .7 


22 

53 

23 .6 

Antares 

June 1st 


-26 15.6 


16 

24 

40.5 



MOON, 1922 







MARCH 9th 





G. M. T. 

Right Ascension 

Declination 


S. D. 


H. P. 

Hrs. 








6 

7h 49m 9s 

297 

+15° 58'.3 

115 

16'.3 


59'.9 

8 

7 54 6 

297 

15 46.8 

120 

16 .4 


59.9 

10 

7 59 3 

298 

15 34.8 

124 

16 .4 


60 .0 

12 

8 4 1 

297 

15 22.4 

128 

16 .4 


60.1 









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